Bargaining and Pricing in Networked Economic Systems

University of Pennsylvania ScholarlyCommons

Publicly Accessible Penn Dissertations

Fall 2011

Tanmoy Chakraborty University of Pennsylvania, [email protected]

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Recommended Citation Chakraborty, Tanmoy, "Bargaining and Pricing in Networked Economic Systems" (2011). Publicly Accessible Penn Dissertations. 445. https://repository.upenn.edu/edissertations/445

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/445 For more information, please contact [email protected]. Bargaining and Pricing in Networked Economic Systems

Abstract Economic systems can often be modeled as games involving several agents or players who act according to their own individual interests. Our goal is to understand how various features of an economic system affect its outcomes, and what may be the best strategy for an individual agent.

In this work, we model an economic system as a combination of many bilateral economic opportunities, such as that between a buyer and a seller. The transactions are complicated by the existence of many economic opportunities, and the influence they have on each other. For example, there may be several prospective sellers and buyers for the same item, with possibly differing costs and values. Such a system may be modeled by a network, where the nodes represent players and the edges represent opportunities. We study the effect of network structure on the outcome of bargaining among players, through theoretical modeling of rational agents as well as human subject experiments, when cost and values are public information.

The interactions get much more complex when sellers' cost and buyers' valuations are private. We design and analyze revenue maximizing strategies for a seller in the presence of many buyers, when the seller has uncertain information or no information about the buyers' valuations. We focus on developing pricing strategies, and compare their performance against truthful auctions. We also analyze trading strategies in financial markets, where a player quotes both buying and selling prices, again with uncertain or no information about future price evolution of the financial instrument.

Degree Type Dissertation

Degree Name Doctor of Philosophy (PhD)

Graduate Group Computer and Information Science

First Advisor Michael Kearns

Second Advisor Sanjeev Khanna

Keywords Theory and Algorithms, Computational Economics, Algorithmic Game Theory, Optimization, Computational Finance, Network Economics

Subject Categories Behavioral Economics | Economic Theory | Finance | Theory and Algorithms

This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/445 BARGAINING AND PRICING IN NETWORKED ECONOMIC SYSTEMS Tanmoy Chakraborty

A DISSERTATION in Computer and Information Science Presented to the Faculties of the University of Pennsylvania in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy 2011

Supervisor of Dissertation Co-Supervisor Signature Signature Michael Kearns, Professor Sanjeev Khanna, Professor Computer and Information Science Computer and Information Science

Graduate Group Chairperson Signature Jianbo Shi, Professor, Computer and Information Science

Dissertation Committee Sudipto Guha, Professor, Computer and Information Science Ali Jadbabaie, Professor, Electrical and Systems Engineering Sampath Kannan, Professor, Computer and Information Science Eva Tardos, Professor, Computer Science, Cornell University (External) ABSTRACT

BARGAINING AND PRICING IN NETWORKED ECONOMIC SYSTEMS Tanmoy Chakraborty Supervisors: Michael Kearns and Sanjeev Khanna

Economic systems can often be modeled as games involving several agents or players who act according to their own individual interests. Our goal is to understand how various features of an economic system affect its outcomes, and what may be the best strategy for an individual agent. In this work, we model an economic system as a combination of many bilateral economic opportunities, such as that between a buyer and a seller. The transactions are complicated by the existence of many economic opportunities, and the influence they have on each other. For example, there may be several prospective sellers and buyers for the same item, with possibly differing costs and values. Such a system may be modeled by a network, where the nodes represent players and the edges represent opportunities. We study the effect of network structure on the outcome of bargaining among players, through theoretical modeling of rational agents as well as human subject experiments, when cost and values are public information. The interactions get much more complex when sellers’ cost and buyers’ valuations are private. We design and analyze revenue maximizing strategies for a seller in the presence of many buyers, when the seller has uncertain information or no information about the buyers’ valuations. We focus on developing pricing strategies, and compare their performance against truthful auctions. We also analyze trading strategies in financial markets, where a player quotes both buying and selling prices, again with uncertain or no information about future price evolution of the financial instrument.

ii Contents

1 Introduction 1 1.1 ExpectationsfromaGoodModel...... 2 1.2 Networked Bargaining ...... 4 1.2.1 BehavioralExperiments ...... 5 1.2.2 Theoretical Modeling and Non-Linear Utility ...... 6 1.3 MechanismDesignandPricing ...... 7 1.3.1 Limited Information Setting ...... 7 1.3.2 Bayesian Information Setting ...... 8 1.4 Pricing Strategies for a Financial Market Maker ...... 10 1.5 ConcludingRemarks...... 11

2 Behavioral Study of Networked Bargaining 12 2.1 OverviewofResults ...... 14 2.2 Background ...... 15 2.3 ExperimentalDesign...... 18 2.3.1 Irregular Graphs ...... 19 2.3.2 Regular Graphs ...... 21 2.4 SystemOverview...... 22 2.4.1 Human Subject Methodology ...... 24 2.4.2 SessionOverview...... 24 2.5 Results...... 25 2.5.1 Social Welfare ...... 25 2.5.2 Nodal Diﬀerences...... 27

iii 2.5.3 Comparison with Theoretical Models ...... 32 2.5.4 Human Subject Diﬀerences ...... 33 2.6 Conclusion ...... 36

3 Networked Bargaining with Non-Linear Utilities 37 3.1 OverviewoftheModel...... 38 3.1.1 Overview of Results and Techniques ...... 41 3.1.2 Related Work ...... 43 3.2 Preliminaries ...... 45 3.2.1 Proportional Bargaining Solution (PBS) ...... 46 3.2.2 Nash Bargaining Solution (NBS) ...... 46 3.2.3 Stability and Equilibrium ...... 48 3.2.4 Bargaining Concepts as Nash Equilibria ...... 48 3.3 Linear Utility Functions: Characterizing All Equilibria ...... 49 3.4 Existence of Equilibrium for Non-Linear Utility Functions ...... 50 3.5 The Bargain Monotonicity Condition ...... 52 3.6 ComputingEquilibria ...... 55 3.6.1 Algorithmic Results ...... 55 3.6.2 The Algorithm ...... 57 3.7 Inequality of Sharing in PBS vs NBS equilibria ...... 59 3.7.1 Inequality in NBS equilibrium ...... 59 3.7.2 InequalityinPBSequilibrium...... 61 3.8 SimulationStudies ...... 62 3.8.1 Methodology ...... 63 3.8.2 Correlation Between Degree and Wealth ...... 64 3.8.3 Regression Coeﬃcients...... 64 3.8.4 Division of Wealth on Edges ...... 65 3.8.5 Other Utility Functions ...... 66 3.9 UniquenessofEquilibrium...... 68 3.9.1 PBS Equilibrium is Not Unique ...... 68 3.9.2 NBS Equilibrium is Not Unique ...... 69

iv 3.9.3 Uniqueness in Regular Graphs ...... 69 3.9.4 Characterization of Bargain Rallying Condition ...... 70 3.10 Generalized Utility Functions ...... 72 3.11 Conclusion ...... 73

4 Item Pricing in Combinatorial Auctions 74 4.1 Preliminaries ...... 82 4.1.1 Notation ...... 84 4.1.2 Optimal Social Welfare and Revenue Approximation ...... 85 4.1.3 The Single Buyer Setting with Uniform Pricing Strategies ...... 86 4.1.4 Optimizing with Unknown Parameters ...... 88 4.2 Improved Lower Bounds for Static Uniform Pricing ...... 89 4.2.1 A Hard Two-Player Instance ...... 90 4.2.2 Extensions of the Two-Player Instance ...... 93 4.3 Dynamic Uniform Pricing Strategies ...... 95 4.3.1 A Dynamic Uniform Pricing Algorithm ...... 95 4.3.2 Lower Bound for Dynamic Uniform Pricing ...... 99 4.3.3 Dynamic Monotone Uniform Pricing Strategies ...... 103 4.4 Static Non-Uniform Pricing ...... 107 4.4.1 Full Information Setting ...... 107 4.4.2 Buyers with -XOS Valuations ...... 108 4.5 Conclusion ...... 111

5 Sequential Posted Pricing in Multi-Unit Auctions 113 5.1 Preliminaries ...... 116 5.1.1 Basic Result ...... 118 5.2 LP-based Algorithm for Large K ...... 119 5.2.1 Approximation Factor ...... 122 5.3 PTAS for constant K ...... 124 5.3.1 PTAS for Computing SPM ...... 125 5.3.2 PTAS for Computing ASPM ...... 129

v 5.4 Additional Details ...... 133 5.4.1 Discretization ...... 133 5.4.2 A Property of Poisson Distribution ...... 134

6 Market Making and Mean Reversion 135 6.1 A General Characterization ...... 140 6.2 Mean Reversion Models ...... 143 6.2.1 Ornstein-Uhlenbeck Processes ...... 145 6.2.2 The Schwartz Model ...... 148 6.3 TradingFrequency ...... 150 6.4 Conclusion ...... 152

7 Summary of Results 153 7.1 Behavioral Study of Networked Bargaining ...... 153 7.2 Networked Bargaining with Non-Linear Utilities ...... 154 7.3 Item Pricing in Combinatorial Auctions ...... 154 7.4 Sequential Posted Pricing in Multi-Unit Auctions ...... 155 7.5 Market Making and Mean Reversion ...... 155

8 Future Research Directions 156 8.1 Networked Bargaining with Incomplete Information ...... 156 8.2 Bundle Pricing for Multi-Parameter Mechanism Design ...... 157 8.3 Mechanism Design for Sellers with Non-Linear Utility ...... 157 8.4 Models of Financial Markets and Optimal Strategy for Market Makers . . . 158

vi List of Tables

2.1 External demand imbalances shape bargaining results ...... 31 2.2 Comparison between no-cost and zero-cost experiments ...... 35

4.1 Summary of results on item pricing in combinatorial auctions ...... 79

vii List of Illustrations

2.1 Representativesofnetworksusedinexperiments ...... 19 2.2 Diamond torus and cycle-with-chords networks ...... 22 2.3 Networks with global imbalance ...... 23 2.4 Screenshot of player’s interface for bargaining ...... 24 2.5 Eﬃciency of greedy algorithm versus human subjects ...... 26 2.6 Histogram of deal splits ...... 28 2.7 Average inequality value versus Social Eﬃciency ...... 29 2.8 People with patience win bigger splits ...... 34

3.1 PBS and NBS regression coeﬃcients ...... 65 3.2 Inequality in PBS and NBS equilibria ...... 66 3.3 Inequality and degrees of endpoints ...... 67 3.4 Inequality for diﬀerentutilityfunctions...... 67

6.1 Proof by picture: Matched and unmatched trades ...... 143 6.2 Growth of profit with time ...... 151 6.3 Average profit versus trading frequency ...... 151 6.4 Standard deviation of profit versus trading frequency ...... 151

viii Chapter 1

Introduction

Economic interactions are ubiquitous. They happen whenever we visit a store, physical or online, to buy items that range from our regular grocery needs to the latest electronic gadgets. They also include trading on equities exchanges, auctions for rare goods such as the wireless spectrum and auctions for dynamically generated goods such as internet advertisement space. They also happen when an employer hires an employee to work at a mutually agreed salary. These are but only a few examples of economic interactions. The most commonly occurring interactions involve the seller posting a price for each item, and a buyer has to make a choice of whether to buy the item at the posted price. Often sellers announce more complex pricing schemes, such as oﬀering discounts on bulk purchases. In contrast, in many auctions, such as that of art and antiques, internet advertisement space and wireless spectrum, the buyer quotes the maximum price he is willing to pay, known as a bid, and the seller chooses to sell the item to one of the buyers based on the bids received. Prices are negotiable in some markets, where a buyer can come back with a quote that the seller may accept or reject, or come back with yet another quote. Such bargaining commonly occurs when the buyers and sellers are in live communication during the economic interaction. For example, prices are often negotiable in small shops, where the shopkeeper has the authority to alter prices as he wishes. Similarly, prices are negotiated when one is looking to buy or even rent a house, or when a football or baseball player negotiates and signs a contract with a club. The goal of this work is to understand the outcomes of interactions and strategies of

1 players in various economic systems. We can model these economic systems as a combination of multiple bilateral economic opportunities, each of which may yield a surplus or profit to the players involved, depending on their actions. To elaborate, consider a prospective seller who holds an item that was perhaps acquired at a certain cost, and a prospective buyer who has a certain value for that item. If the seller and the buyer can agree on transfer of the item at a negotiated price, then the surplus generated is the valuation of the buyer minus the cost for the seller. The buyer receives a share of the surplus that is equal to its valuation minus the price, while the seller makes a profit equal to price minus cost. In all the above examples of economic systems, one can identify buyers and sellers, and represent every possible pair of agents who can trade between themselves (on an item of common interest) as an economic opportunity. We can model an economic system as a game, where the payoffs to the agents are a combination of the profits realized from the economic opportunities they are involved in. The exact structure of the payoff of a player can be simply the sum of payoffs obtained from the different opportunities, or it can be a more complex non-linear function of these payoffs, as seen in the well-known phenomenon of diminishing marginal utility. It can even involve parameters such as some measure of fairness in comparison to the fortunes of other players and social welfare. We can then consider equilibria of such a game as a reasonable outcome of the economic system. There are many sellers and many buyers for the same or similar items, with possibly differing costs and valuations. Further, the market place can have multiple distinct items, and a buyer or seller may be interested in trading multiple items. Payoff of a player can be a complex function; further, the set of feasible actions is often constrained by the inventory size of a seller or the budget of a buyer. So outcome of an economic interaction is influenced by other economic opportunities too, and we focus on understanding these mutual effects.

1.1 Expectations from a Good Model

Modeling every feature of a complex economic system is often impossible, since many features can hardly be measured. Modeling a player’s responses to various situations is also hard to model precisely, since it varies over time and the human components in a decision

2 process vary from person to person. So we often impose some intuitive rationality on the players. We assume that the players can find a better strategy when there exists one, and analyze the eventual outcome of the system under this assumption. The goal of modeling players’ behavior and parameters of the economic system is to obtain an accurate enough model so that the outcomes in the model match those in the real world. At the same time, the model of player’s behavior should be realistic, that is, it should intuitively match what players might actually do. A somewhat simpler but related objective of a model can be to propose an outcome that can be deemed to be fair to the participants given their resources and capabilities. This notion of fairness often needs to be guided by what happens in the real world. Often, for simplicity, players are assumed to be always capable of identifying the best strategy, and Nash equilibrium is proposed as a solution concept. This assumption often puts the onus of an extremely difficult decision on individual players. This is especially true for games played over a long period of time, since the action space of the player grows exponentially with time. This contradicts our common intuition that an average buyer does not solve complex optimization problems but instead takes simple, and often myopic, steps that appear to be beneficial. Moreover, the optimization problems are often so complex that it is difficult to even design algorithms that will run sufficiently fast on modern computers, so even automated agents with significant computational power may not be able to identify their optimal strategy. So it is not surprising that the equilibrium outcomes often do not match with real world outcomes. Further, when the model has many equilibria, its value as a predictive model gets severely diluted. A complex model also has a negative impact on theoretical analysis, since we are less likely to find succinctly characterized equilibrium. Further, finding an optimal strategy or an equilibrium may become computationally intractable. In such a situation, the model would hardly provide any useful insight. If we are not even able to simulate the model and observe what an equilibrium may look like, what purpose would the model serve? Thus, a fairly basic necessity of a good model is that it should be tractable – players should have simple enough decisions to make, and the outcome of the system, which we shall refer to as a solution concept, should also be easy to compute.

3 To complicate matters, the amount of information available to the players have a strong effect on the outcomes. For example, if both the buyer and the seller are aware of each other’s cost and valuation, then they both know that they are simply negotiating on how to share the mutually known surplus. If the seller knows his cost but not the buyer’s valuation, the basic notions of defining equilibrium start to crumble, since the seller’s best strategy is not well-defined. Such a game is often referred to as an incomplete information game, and a standard solution concept here is the Bayes-Nash equilibrium, where it is assumed that each player has Bayesian information or belief, which is a probability distribution, over all parameters that are relevant but unknown to it. In a game played over time, each player gathers information about the unknown parameters by observing actions of other players. Of course, this adds heavily to the complexity of the decision that an individual has to make, and also makes it necessary to model the distributions. Further, it is commonly observed in many simple systems that there are too many Bayes-Nash equilibria, which are also hard to compute. An alternate concept rose in the context of mechanism design. In a model involving a single seller and several buyers, a seller is restricted to choosing a strategy (an auctioning rule) so that the best response of the buyers is to simply communicate their true valuation to the seller, irrespective of the information available to the buyers. Such auctioning rules are referred to as truthful or incentive compatible mechanisms. The design of truthful mechanisms is popular because only the seller has to make a complex decision, and can then convince the buyers that they should simply report their true valuation. So it can be viewed as restriction to a natural class of equilibria. Again, tractability of a model remains crucial to its usefulness, so we shall keep an eye on the computability of revenue or welfare maximizing mechanisms. With this background information on what we expect from a theoretical model of an economic system, we now proceed to describe the models that are presented in this work.

1.2 Networked Bargaining

Along the lines of the discussion above, an economic system can be modeled by a network, where the nodes represent players and the edges represent economic opportunities. We

4 study a networked bargaining game, where a population of players simultaneously negotiate bilateral opportunities that yield a known surplus if the two players agree on how to share it – we shall refer to this agreement as signing of a deal. We also place a limit on each node, which is the maximum number of deals that the player at that node can sign. This models the demand volume of a buyer or the inventory size of a seller, and induces competition among the players.

1.2.1 Behavioral Experiments

In Chapter 2, we report on a series of highly controlled human subject experiments in networked bargaining. Results in this chapter were published in collaboration with Stephen Judd, Michael Kearns and Jinsong Tan [23]. Behavioral experiments allow us to design toy economies with different structures that can test our hypothesis – in contrast, we often get a single or few snapshots of a real economic system, with little data to make direct comparisons, or create and test the accuracy of a predictive model. They also provide us with a direct insight into the collective behavior of human players under various economic tensions. We designed networks with up to 36 nodes, and recruited human subjects to play these games in a laboratory, where the only communication occurred through computer interfaces designed by us. The basic interaction between two players is the decision of how to share a mutual payment; the players simultaneously negotiate on all incident edges by sending offers and counter-offers on each edge. Various theories predict, to different levels of uniqueness, what the shares will be. We analyze our experimental results from three points of view: social efficiency, nodal differences, and human differences; and contrast our behavioral results with the theories. Our experiments show that human subjects can consistently find solutions that are so- cially more efficient than random solutions, but loses this advantage due to local disagree- ments that lead to under-consumption of resources. We also found that the bargaining power of a node, measured by the average size of shares received, is enhanced by high degree and low limit, while it is also enhanced by low degree and high limit of its neighbors. Further, we found that patient players tends to get larger shares on average. These are only

5 some of our ﬁndings – a detailed exposition can be found in Chapter 2.

1.2.2 Theoretical Modeling and Non-Linear Utility

We also study networked bargaining through theoretical modeling of rational agents. Our model extends any two-player bargaining solution concept, such as Nash and proportional bargaining solutions, to an equilibrium over the networked game. The model is based on a local equilibrium concept introduced by Cook and Yamagishi [31]. Players are modeled as making myopic adjustments to the deals, using local but dynamic information about their neighbors’ accumulated wealth, till every deal appears to be fair given the local information. Such a model can be viewed as bounded rationality applied to networked bargaining, and models buyers as simple decision-makers. We focus on a setting where no limit on the maximum number of signed deals is imposed on the players. In this setting, there is no network effect on the outcomes when players have linear utility, that is, the economic opportunities have no effect on each other. However, we show that non-linear utility causes network structure to have an effect on the outcome. This may be justified by the following intuition: with no demand or supply constraint, one may expect the bilateral negotiations to become independent. However, non-linear utility continues to bind them together, since a player’s marginal utility from a deal may change depending on the amount earned from other deals. We show that equilibrium exists in all networks, and is unique in most networks, but not all. As a consequence of diminishing marginal utility, a node’s bargaining power is enhanced if it has high degree, its neighbors have low degree, their neighbors have high degree, and so on. This can be summarized as a ”rich-get-richer” effect. Though players only take local myopic decisions, the effect of network structure propagates beyond the immediate neighborhood. Proportional bargaining proposes solutions with greater inequality than Nash bargaining. We also propose a natural algorithm to compute an equilibrium, based on asynchronous selfish updates. The algorithm converges quickly in general, and we prove that a specific variant converges fast on all bipartite networks. These results are presented in Chapter 3. Results in this chapter were published in collaboration with Michael Kearns and Sanjeev Khanna [24, 26].

6 1.3 Mechanism Design and Pricing

In the study of networked bargaining described above, the surplus on each edge was mutually known to the players involved. If the buyers’ valuations and the sellers’ costs are private information, as they often are, then the problem becomes much harder to model – first because simple modeling as an equilibria gives too many outcomes that are difficult to compute, and second because it is hard to measure the Bayesian information that each player is using. Studying such incomplete information bargaining has thus been largely restricted to a single pair of buyer and seller, and even then obtaining structural results or efficient computation has required a lot of effort – [35] provides a survey on incomplete information bargaining. A more popular path has been to study the notion of truthful mechanisms, and this model incorporates a single seller and multiple buyers. Recently, extensions to models with more than one seller has been suggested [78], but understanding of such models is very limited. In this work, we restrict ourselves to the one seller, many buyers setting, leaving the general networked setting as a suggestion for future work. As mentioned before, one of the most frequently occurring economic interaction involve the seller setting prices without (precise) knowledge of the buyer’s valuations, and the buyer making a decision of which items to buy based on the set prices. This is what we typically face when buying from a large grocery or departmental store, or from Amazon’s online store. It is not difficult to see that since the prices are set independent of the buyers’ valuations, buyers have no incentive to lie about their valuations, that is, pricing mechanisms are truthful. We focus on designing revenue maximizing pricing strategies for the seller. We also examine the gap in revenue between an optimal pricing strategy and an optimal auction (the latter always performs better than the former), and find that it is small in some natural settings.

1.3.1 Limited Information Setting

We may assume that the seller has no information about the buyers’ valuations and tries to perform well against whatever valuation they may have – we study this setting in Chapter 4. Results in this chapter were published in collaboration with Zhiyi Huang and Sanjeev Khanna [22]. We consider the Item Pricing problem for revenue maximization, where a

7 single seller with multiple distinct items caters to multiple buyers with unknown subadditive valuation functions who arrive in a sequence. Subadditive valuation is a natural extension of diminishing marginal utility to a multi-dimensional domain. The seller sets the prices on individual items, and we design randomized pricing strategies to maximize expected revenue. We consider dynamic uniform strategies, which can change the price upon the arrival of each buyer but the price on all unsold items is the same at all times, and static non- uniform strategies, which can assign different prices to different items but can never change it after setting it initially. Dynamic strategies can be especially useful in online stores, where it is easy to show different prices to different buyers. We design pricing strategies that guarantee poly-logarithmic (in number of items) approximation to maximum possible social welfare, which is an upper bound on the maximum revenue that can be obtained by any pricing strategy or any individually rational mechanism. We also show that any static uniform pricing strategy cannot yield such approximation, thus highlighting a large gap between the powers of dynamic and static pricing. Finally, our pricing strategies imply polylogarithmic approximation for revenue-optimal incentive compatible mechanisms, in multi- parameter settings with subaddititve valuations. Even with Bayesian information about buyers’ valuations, no computationally efficient mechanism with a better approximation factor is known for such general valuation functions.

1.3.2 Bayesian Information Setting

In the complete absence of information, it is difficult to even define an optimal mechanism. In fact, it is unlikely that we will get anything better than logarithmic approximation when competing with a strategy that knows the valuation functions. Alternately, we can assume that the seller has Bayesian information about the buyers’ valuations, and this allows us to define an optimal mechanism in terms of expected revenue, where expectation is measured with respect to the Bayesian distributions. Myerson [76] designed optimal mechanism for single parameter problems, where each buyer is single-minded, that is, his valuation is a fixed positive value for a particular set of items or service, and it is zero if this particular set is not allocated to him. A simple setting where this is applicable is multi-unit auction

8 of a single type of good, where each buyer has interest in at most one copy of the good. However, Myerson’s optimal mechanism is not a pricing mechanism, and asks for bids from buyers before deciding their payment. So it is applicable in auction settings where buyers are asked to bid, but not in general stores where pricing is the standard norm. Further, buyers may find it difficult to precisely measure and quote their value for an item – on the other hand, pricing simply asks buyers to make a binary decision – whether their value is above the quoted price or not. Since buyers often uses less computational resources than sellers, this gives a practical advantage. In Chapter 5, we focus on designing near- optimal pricing mechanisms for multi-unit settings in a Bayesian environment. Results in this chapter were published in collaboration with Eyal Even-Dar, Sudipto Guha, Yishay Mansour and S Muthukrishnan [20]. We design polynomial time algorithms for computing approximately revenue-maximizing sequential posted-pricing mechanisms (SPM) in K-unit auctions. A seller has K identical copies of an item to sell, and there are n buyers, each interested in at most one copy, who have some value for the item. The seller does not know the values of the buyers, but knows the distributions that they are drawn from. Each buyer’s value is drawn independently from a distribution associated with that buyer. In an SPM, the seller sets a price for each buyer (without looking at any bid but only using the distributional information), the buyers are approached in a sequence chosen by the seller, and a buyer buys the item if its value exceeds the price posted to it. An SPM specifies the ordering of buyers and the posted prices, and may be adaptive or non-adaptive in its behavior. A non-adaptive SPM is also equivalent to offering prices to the buyers in parallel, and choosing to sell the items to up to K of the buyers who accept their respective offered prices. Our first algorithm computes prices using a linear program, and we show that if the buyers are approached in decreasing order of prices, the the expected revenue of the computed

K SPM is at least (1 K ) (1 1 ) times that of Myerson’s optimal mechanism (which − K!eK ≈ − √2πK is an upper bound on any pricing strategy). Thus the gap between an optimal SPM and an optimal auction vanishes solely as a function of the inventory size is increased, and this result holds even if the buyers’ pool is chosen adversarially based on K. In other words, a seller who owns a large inventory may commit to use a pricing strategy instead of an

9 auction, and be assured that this commitment can lead only to a small regret, even before gathering any knowledge about the buyers’ pool. For constant K, there is at least a constant multiplicative gap between pricing and auction. We design polynomial time approximation schemes for the optimal non-adaptive and adaptive SPM respectively.

1.4 Pricing Strategies for a Financial Market Maker

In Chapter 6, we study the profitability of market making in price time series models. Results in this chapter were published in collaboration with Michael Kearns [25]. In financial markets, every player can simultaneously act as both a buyer and a seller, buying and selling stocks by participating in a continuous two-sided auction at an electronic exchange (we assume that a player can short a stock). In a two-sided auction, a player may quote buy prices and/or sell prices, and a trade is executed if a buy and sell order can be matched. Market making refers broadly to trading strategies that seek to profit by providing liquidity to other traders, while avoiding accumulating a large net position in a stock. In this chapter, we study the profitability of market making strategies in a variety of time series models for the evolution of a stock’s price. Typically, a market maker quotes both a buy and a sell price for a financial instrument or commodity, hoping to profit by exploiting the difference between the two prices, known as the spread. Intuitively, a market maker wishes to buy and sell equal volumes of the instrument (or commodity), and profit from the difference between the selling and buying prices. Market makers are common in foreign exchange trading, where most trading firms offer both buying and selling rates for a currency. They also play a major role in equities exchanges, and exchanges often appoint trading firms to act as official market makers for equities. We first provide a precise theoretical characterization of the profitability of a simple and natural market making algorithm in the absence of any stochastic assumptions on price evolution. This characterization exhibits a trade-off between the positive effect of local price fluctuations and the negative effect of net price change. We then use this general characterization to prove that market making is often profitable on mean reverting time series – time series with a tendency to revert to a long-term average. Mean reversion has

10 been empirically observed in many markets, especially foreign exchange and commodities. We show that the slightest mean reversion yields positive expected profit, and also obtain stronger profit guarantees for a canonical stochastic mean reverting process, known as the Ornstein-Uhlenbeck (OU) process, as well as other stochastic mean reverting series studied in the finance literature. We also show that market making remains profitable in expectation for the OU process even if some realistic restrictions on trading frequency are placed on the market maker.

1.5 Concluding Remarks

After completing a detailed exposition of the models studied in this thesis and the various results, we revisit and summarize them in Chapter 7. In Chapter 7, we provide a clear and detailed contrast of our contributions with previously known work, which serves to summarize the specific contributions of this thesis to the area of computational economics. We also outline some work in progress, open problems and future research directions in the final Chapter 8 that are inspired from the models and results described above. Briefly, they include

Study of networked bargaining with incomplete information, through theoretical mod- • eling as well as behavioral experiments.

Designing eﬃcient algorithms to compute pricing strategies that price bundles of items • as well as single items, and achieve provably greater revenue in the process.

Analyzing the eﬀect of diminishing marginal utility on the optimal pricing strategy of • a single seller, and computing utility maximizing pricing strategies and auctions. In a Bayesian setting, diminishing marginal utility is a standard model for capturing risk aversion. I have been involved in some recent progress on this topic [11].

Developing more realistic models for a ﬁnancial market, and the design and analysis • of pricing strategies for a market maker in appropriate market models. This is also a work in progress.

11 Chapter 2

Behavioral Study of Networked Bargaining

In recent years there has been much research on network-based models in game theory. Topics of attention include the effects of network topology on the properties of equilibrium in social and economic networks, the price of decentralized decision-making in networking problems like routing, game-theoretic models of network formation, convergence to equilibrium and computability of equilibrium in networked games, and many others. This large and growing literature has been almost exclusively theoretical, with few accompanying empirical or behavioral studies examining the relevance of the mathematical models to actual behavior. In this chapter, we report on a series of highly controlled human subject experiments in networked bargaining. Results in this chapter were published in collaboration with Stephen Judd, Michael Kearns and Jinsong Tan [23]. Realization of an economic opportunity and the surplus associated with it is usually contingent upon agreement between participants on how to split the surplus. When multiple economic opportunities are negotiated simultaneously, they can affect each other. Even when only local information is available, the effects of one transaction can spread beyond its immediate neighborhood. We study the nature of these interactions through behavioral experiments with human subjects. Specifically, we study a networked bargaining game, where a population of players simultaneously negotiate on multiple bilateral economic opportunities. Recall that a bilateral

12 economic opportunity can be, for example, a relation between a buyer and a seller for a particular item, or it can be an opportunity to run a business as partners. Moreover, these opportunities may not exist between all pairs of players. For example, items are heteroge- nous, and an economic opportunity exists only between a buyer interested in a particular item and a seller who possesses that item in its inventory. It is convenient to represent the players as nodes of a network, where each opportunity is represented by an edge between two nodes, and interaction between transactions can be captured by the effect of network structure on the outcome. We shall refer to such a network as a bargaining network, each edge or transaction as a deal, the participants in the transaction as endpoints of the deal, and an agreement between its endpoint nodes to share the surplus as closing of the deal. In our networked bargaining experiments, we impose an exogenous network of economic opportunities and assign human subjects to these nodes as players of the game. It is also natural for players to have a limit or budget on the volume of transaction they can involve themselves in. For example, a seller may have a limit on the inventory which is less than the demand volume of potential buyers. A buyer too may have a consumption capacity that is less than the total supply volume of prospective sellers of the particular product. We impose a limit on each node, which is the number of deals that a player at that node is allowed to close. This limit is often less than the degree (number of incident edges) of the node, and cannot be more than the latter. From a game-theoretic point of view, the imposition of limits create competition between nodes to grab the attention of their common neighbors. We were partly inspired by a long line of previous theoretical work among sociologists, economists and computer scientists on networked bargaining games with limited exchange, which proposed solution concepts and analyzed the effect of network topology on properties such as bargaining power of a node and efficiency [31, 51, 14, 74, 17, 68, 4, 24, 26, 34, 27, 1, 2, 73]. A notable feature of these theories is the prediction that there may be significant local variation in splits purely as a result of the imposed deal limits and structural asymmetries in the network. One can view our experiments as a test of human subjects’ actual behavior at this game in a distributed setting using only local information. Our experiments are among the largest behavioral experiments on network effects in bargaining conducted to

13 date. We adopt many of the practices of behavioral game theory, which has tended to focus on two-player or small-population games rather than larger networked settings. In each of our experiments, three dozen human subjects simultaneously engage in one-to-one bargaining with partners deﬁned by an exogenously imposed network. Our work continues a broader line of research in behavioral games on networks at the University of Pennsylvania [67, 64, 66].

2.1 Overview of Results

In an extensive and diverse series of behavioral experiments, and the analysis of the resulting data, we address a wide range of fundamental questions, including: the relationships between degree, deal limits, and wealth; the effects of network topology on collective and individual performance; the effects of degree and deal limits on various notions of “bargaining power”; notions of “fairness” in deal splits; and many other topics. The networks used are inspired from common models in social network theory, including preferential attachment graphs, and some specifically-tailored structures. In all our experiments, the number of deals that were closed was above 85% of the maximum possible number. This is high enough to demonstrate real engagement, and low enough to demonstrate real tension in the designs. Most of the deeper findings can be related to existing network bargaining theory. Al- though deals are often struck with unequal shares, more than one-third of the deals are equally shared, thus indicating that people, while behaving as self-interested actors, also have an aversion towards inequality. Network topologies have enough of an effect that they can be distinguished statistically via individual wealth levels and other measures. Higher degree, for example, tends to raise bargaining power while higher deal limits tend to decrease it. But while local topology affects bargains, invisible competition also affects it, even when the local topologies are indistinguishable. We find the expected effects of higher deal limits in the first neighborhood and higher degrees in the first and second neighborhoods, but neither degree distribution nor deal limit distribution is sufficient to determine the inequality of splits. In summary,

14 there is a rich interaction between network and wealth that needs more study. Other ﬁndings that speak to no existing theories but might provoke some new ones are the following:

There is a positive correlation between inequality and social efficiency. • Failures to agree on a split (as opposed to failures to find the best global trade con- • figuration) form the greater part of missing efficiency. We term this loss the price of obstinacy.

In a few of our experiments, we imposed a cost on closing a deal. We found that social • eﬃciency was higher when some uncertainty existed about a partner’s costs.

Finally, there are two curios that seem more about psychological dynamics than economics:

People who are patient bargainers tend to make more money. • For a closed deal, we can identify the proposer proposer as the player who made the • last offer on the deal, so that its offer was accepted by the other player (the acceptor). The share of the proposer stochastically dominates the share of the acceptor, with high statistical significance. In hindsight, we can provide the following explanation: a player is willing to accept an offer that is few cents lower than what it deems fair, but is unlikely to propose such an offer.

In the ensuing sections, we review relevant networked bargaining theories, describe our experimental design and system, and present our results.

2.2 Background

Networked bargaining with deal limits on the nodes, also known in the sociology literature as networked exchange with substitutable or negatively connected relations (eg. [17]), has been studied for decades. Several theoretical models have been designed to predict or propose how wealth should be divided [51, 74, 31, 83], and human subject experiments have been conducted on a few small graphs (up to 6 nodes) [32, 33, 83], albeit with diﬀerent

15 interfaces and mechanisms than ours. Some of the theoretical models are based on limited experimentation, along with simulated human behavior on slightly larger graphs [33]. A few models are based strongly on notions of game-theoretic rationality and are natural extensions of standard economic literature to social networks. Two models that belong to this class were introduced by Cook and Yamagishi [31] and by Braun and Gautschi [17]. We shall mainly focus on these two models. The model given by Cook and Yamagishi, sometimes referred to as equidependence theory, is one of the most recognized theoretical model, that has received a lot of recent focus from the theoretical computer science community [68, 4]. It proposes an equilibrium concept based on players using local (both static and dynamic) information only, and it is hte main theoretical model that inspired our experiments. Though Cook and Yamagishi[31] considered only unique exchange networks (that is, where each vertex may close only a single deal), the model is easily extendable to networks with varying deal limits. Every node is assumed to play strategically with selfish game-theoretic rationality. The outside option of a node is defined as the highest offer it can rationally receive from any of its neighbors, such that closing that deal would benefit both parties, compared to the given state. An outcome is said to be stable if every player’s earning is more than its outside option. Game-theoretic rationale suggests that an outcome should be stable if the players act in a myopically selfish manner. Cook and Yamagishi propose that the achieved outcome must be stable; moreover, they propose that the achieved outcome should be balanced, that is, two parties that close a deal should have equal additional benefit from this edge, where additional benefit is measured as the amount by which the earning of a player exceeds its outside option. Kleinberg and Tardos [68] showed that a stable and balanced outcome exists on all bipartite networks, but may not exist in all networks, and if it does, the closed deals in a stable outcome form a maximum matching. This equal division of surplus is stipulated by standard two-player bargaining solutions such as the Nash Bargaining Solution and Proportional Bargaining Solution, for players with linear utilities [77, 15]. Recently, we analyzed the effect of diminishing marginal utility, in the setting where there are no limits on the number of deals [26] – this analysis is presented as a part of this work in Chapter 3. In this case, the model predicts that all deals should be closed, and if players have linear

16 utility (which is assumed in the Cook-Yamagishi model), all deals should be shared equally. Unequal splits may occur only if players have non-linear utility. Though a balanced outcome seems to be the most robust theoretical model, it has several drawbacks as a predictive model. First and foremost that it does not exist on even simple networks such as a triangle; and when it exists, there is a balanced outcome for every maximum matching in the network. This makes it computationally hard to even enumerate all the balanced outcomes in a network, and non-uniqueness reduces the predictive value of such a model. Another drawback is that the model often suggests that some edges will be shared so that one party gets an infinitesimal share, and the other party gets practically the entire amount. For example, a node that has at least two leaves (nodes of degree 1) as neighbors always ends up with maximum possible profit, due to competition between the leaves. However, even previous small-scale experiments [83] have suggested that such a phenomenon does not happen, and in our experiments, players rarely close a deal that extremely favors one of the players. Thus, when human subjects are involved, perfect local rationality seems to be an incorrect assumption. A model proposed by Braun and Gautschi [17] defines a “bargaining power function” on nodes that depends only on the degree of the node and degrees of its neighbors. This function increases with increase in degree of the node, and decreases with increase in degrees of its neighbors, and is independent of all other network aspects. On each edge, the division of wealth, if a deal is made, is stipulated to be proportional to the bargaining power of the adjacent nodes. The bargaining power functions do not distinguish between different limits on nodes, and generally assume that relations are negatively connected: that is, for any given player, closing one deal reduces the maximum value that can be obtained from other edges deals. This makes the model quite inadequate as a predictor for our experiments. The other feature of this model is that network effects are quite local in nature, since even slightly distant properties such as the degrees of neighbors of neighbors do not have an effect on the bargaining power function. However, the model attempts to capture the notion that the earning of a player depends positively on its own degree and negatively on the degree of its neighbors. We test this notion on fairly large graphs for the first time, and we also show that the degrees of neighbors of neighbors do affect a node positively. Such alternating

17 effects were predicted in previous theoretical models such as that by Markovsky et. al. [74], which said that odd length paths from a node enhance its earning, while even length paths reduce it. A closely related literature is the study of sequential offer bargaining in networked setting [34, 27, 73, 1, 2], where the surplus shrinks with the passage of time, and the negotiation proceeds through controlled rounds of alternating offers and counter-offers. This is in contrast to our experiments, where either player can make an offer at any time on a deal, and the surplus remains unchanged over time, though there is a deadline at which the game ends. The equilibrium concept studied here is subgame perfect equilibria in a game where players have complete information about the entire network, another distinction from our experiments, in which we only provide local information. This solution concept also suffers from drawbacks as a predictive model similar to those outlined above, such as multiplicity of equilibria. The most significant set of previous experiments were done by Skvoretz and Willer [83], who conducted experiments on 6 small networks (each has at most 6 nodes), with only unit deal limits in 4 of them. They found that some common intuitions held true in those networks. For example, players who have deal limit one and multiple leaves as neighbors gets the bigger fraction of a closed deal, and that this fraction reduces if the limit of the player is raised. Among other results, we test such hypotheses extensively on much larger graphs with much more variance in their degree and limit distributions, and establish these hypotheses with very high statistical significance. Larger graphs also allow us to study the effects of network topology aspects that are more involved than the degree or limit of the player.

2.3 Experimental Design

We designed 18 different experimental scenarios (consisting of specific choices of networks and arrangement of deal limits; each such scenario received 3 trials, for a total of 54 short experiments). These scenarios were based on 8 different graphs with a wide variety of details that is exemplified in Figure 2.1. The sole property they share is that they all have 36 nodes. We are thus casting our experimental nets wide here regarding network topology,

18 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1 a b c d

Figure 2.1: Representatives of all the networks used. (a) the PA graph with positive homophily. (b) a torus; the edges running off the top and bottom denote wrap-around connections, as do those off the sides. (c) a simple cycle. (d) the 2ndHood graph for testing second neighborhood effects. as in much of our previous behavioral work. This section describes all the scenarios, at least at a high level. The networks fall into 2 categories: regular graphs (to isolate and explore the effects of variations in deal limits), and irregular graphs (which contain an assortment of different degrees).

2.3.1 Irregular Graphs

We were interested in how bargaining behavior changes with changes in local network structure, and especially with differences in degree. Out of the huge space of such networks, we chose four. The first three we describe all had a common degree sequence, but differed in the way that nodes of each degree connected to nodes of other degrees. We generated a single degree sequence with a distribution that approximately follows a power law, and used it to build three graphs with different patterns of degree-to-degree profiles. We refer to these graphs as PL (for Power Law) graphs.

Power Law Graphs

Since we suspected that degree might have a large inﬂuence on bargaining power (to be conﬁrmed below), it matters to the success of any node what the degrees are of other nodes they need to bargain with. Hence it was important to manipulate the degrees of neighbors as well.

19 By connecting nodes in different ways, we generated 3 graphs that differ in this manner but have the same basic degree distribution. In PLP (for Power Law Positive, indicating locally positive degree correlation) the high degree nodes are connected to other high-degree nodes. It models a world where nodes of different degrees are segregated from each other. In PLN (for Power Law Negative) the high degree nodes are connected to the low-degree ones; it models a world where the connection-poor are likely to be ‘captivated’ by the connection- rich. And PL0 has them all mixed together to disperse such phenomena. It models a world where nodes mingle freely with other types. With each of the PL graphs above, we used each of the following 3 deal limit schemes to obtain 3 3=9different scenarios. The first is the well-studied unique exchange situation × (uniq): all nodes have deal limit 1. The other 2 are neither unique exchange nor unlimited, but represent two points in another large space of possibilities in between those notions. They are best thought of as having random deal limits drawn uniformly between 1 and the degree of the node. We call them limA and limB, and the difference is just that they are different randomizations.

Identical First Neighborhoods

The final irregular graph was designed specifically to test if structure outside the immediate neighborhood of a node would affect its behavior. The network used for this test has two sets of three identical nodes, which are colored blue and red in Figure 2.1d. Both sets have degree 6, and each of their neighbors have degree 7, so the local neighborhoods are indistinguishable in our GUI views. Any differences in behavior must be due to the second neighborhood or aspects even more distant. The second neighborhood of these two sets of nodes are drastically different; the second neighbors of the red nodes includes the 20 leaves while the second neighbors of the blues does not. This graph helped us identify the effects of second neighborhood when the first neighborhoods of two nodes were identical. We used it only with all nodes having deal limit 1. We refer to this scenario as 2ndHood.

20 2.3.2 Regular Graphs

The 8 remaining scenarios were all based on regular graphs. This allowed us to test effects other than degree, like differing deal limits or large-scale market imbalances. One graph is the cycle shown in Figure 2.1. Four of them are identical tori with different deal limit schemes. Finally, three other graphs were used to observe the effects of a global supply imbalance, and are described in section 2.3.2.

Tori

The 4 tori are topologically the same as the 6 6 torus in Figure 2.1b, and are diﬀerentiated × only through deal limits: Uniform Torus (torUniq): all nodes have deal limit 1. • Checkerboard Torus (torChkb): all white nodes have deal limit 1, the others have deal • limit 3. Torus Rows (torRows): alternating rows have deal limit 1 and deal limit 3. • Torus Diamond (torDiamnd): Some vertices have deal limit 1 and some have deal limit • 3. See Figure 2.2.

Imbalanced Supply Networks

The supply networks are 3 regular graphs which were designed to study the effect of a capacity issue which is not apparent at the node, but becomes apparent when contrasting the deal limits of two groups of nodes. Let the external demand of a group be the sum of deal limits of the nodes in the group minus the maximum number of deals that can be closed within the group. In the supply networks, we defined the groups as the left group and right group as shown in Figure 2.3. All nodes have degree 4. All vertices in the left group have deal limit 2, and all vertices in the right group have deal limit 3. In each network, the right group has two different types of neighbors: those that belong to the right group, and those that belong to the left group. It is their differential treatment of the two types that was of interest. Nodes on the left have only one kind of neighbor; they exist just to set up the market conditions for those on the right.

21 3 1 3 1 3 1

1 3 1 1 1 3 3

3 1 1 3 1 1

1 1 3 1 3 1

3 1 1 3 1 1

1 3 1 1 1 3 a

Figure 2.2: (a) Diamond torus, and (b) CWC (cycle with chords). CWC was used only in preliminary session.

The three graphs share the fact that all deal limits are either 2 or 3, and the ones on the right have both types of neighbours. They are diﬀerent in the ratios of external demands between the left and right groups; in the Undersupplied case the right nodes are somewhat starved for deals (seeking 39 when only 30 could be forthcoming), in Equisupplied they are just balanced, and in Oversupplied they have more oﬀers than they can use.

2.4 System Overview

Experiments were conducted using a distributed networked software system we have designed and built over the past several years for performing a series of behavioral network experiments on diﬀerent games. This section brieﬂy describes the user’s view of that system in our bargaining experiments.

22 # #

(degree,limit) Xdeals:Xdeals (degree,limit) internal edges inter-edges internal edges

15 21

(4,2) 30:39 (4,3) 0 60 24 Undersupplied

12 24

(4,2) 24:24 (4,3) 0 48 48 Equisupplied

10 26 (4,2) 20:14 (4,3) 0 40 64 Oversupplied

Figure 2.3: Metadesigns for networks studying diﬀerential treatment of nodes under three global market conditions. The top line is a template for interpreting the others. Xdeals means external demand.

Like most microeconomic exchange models, the model described in Section 2.2 does not specify an actual temporal mechanism by which bargaining occurs, but of course any behavioral study must choose and implement one. At each moment of our experimental system, and on each edge of the network, each human subject is able to express an offer that is visible to the subject’s neighbor on the other end of the edge. See Figure 2.4. The offer expresses the percentage of the benefit that a player is asking for. When the portions on either end of an edge add up to exactly 100%, one of the players is able to close the deal by pressing a special button. Individuals can always see the offers made to them by their neighbors, as well as some additional information (including the degrees and limits of their neighbors, and the current best offers available to their neighbors). When a deal is closed, or when one of the partners has used up his limit of deals, the relevant edge mechanisms are

23 Figure 2.4: Screenshot of player’s interface for bargaining. frozen and no further action is allowed on them. Every game is stopped after 60 seconds. Any money riding on deals not closed within that time is simply “left on the table”, i.e. the players never get it. All communication takes place exclusively through this bargaining mechanism. Actions of a user are communicated to the central server, where information relevant to that action is recorded and communicated to the terminals of other users.

2.4.1 Human Subject Methodology

Our IRB-approved human subject methodology was similar to that of previous experiments at the University of Pennsylvania [67, 64, 66].

2.4.2 Session Overview

The main experimental session we shall study, which employed the network structure and deal limit scenarios described above, consisted of 3 trials each of the 18 scenarios described

24 above, making 54 experiments in all. Each edge had a payment of $2 available, and in the end approximately $2500 was spent on subject payments. Unless mentioned otherwise, this chapter will always be describing this session of experiments. Prior to this main session, we ran a preliminary set of experiments that employed many but not all of the same network structures, but without any deal limits imposed. Some of these experiments also imposed “transaction costs” on vertices for closed deals. We will mention results from this earlier session and contrast them with those of our main session in a couple of places.

2.5 Results

Our results come under three broad categories. The first is about collective performance and social efficiency. The second category examines questions about the differential fates of nodes, depending on their position in the networks and the deal limits they each had. The third category is about the general performance of humans summarizing behavior across all the games they played. This is an area that no economic theory attempts to cover.

2.5.1 Social Welfare

Humans were quite effective at playing these games, but they paid a surprising price for their refusal to close some deals. To quantify how well humans did on this problem, we implemented a greedy algorithm for comparison. Given a graph and deal limits, it repeatedly draws (uniformly at random) an unclosed deal, both of whose endpoints have not already saturated their deal limit, and closes it, until there are none left. To normalize both the human and greedy systems we divide by the Maximum Social Welfare, which is the maximum number of deals that can close in each network, subject to both topology and deal limits. The social welfare is the number actually closed, and the ratio between this and the max is the social efficiency. The observed efficiencies are rendered by blue dots in Figure 2.5. In 6 of the networks (those below the diagonal), the humans did worse than the greedy algorithm. Full efficiency is rare in both systems. One might view this as the behavioral price of anarchy due to selfish players operating with only local information. The greedy algorithm obtained an average of

25 92.14% of the maximum welfare in our networks. In comparison, human subjects achieved an average social welfare of 92.10% of the maximum welfare when averaged over all 3 trials, a surprisingly similar ﬁgure.

100

96 humans of humans 94 of 92 efficiency 90 efficiency

88 potential 86 86 88 90 92 94 96 98 100 efficiency of greedy algorithm

Figure 2.5: Scatterplot of greedy algorithm eﬃciency versus humans, one dot for each of the 18 scenarios (averaged over trials). Blue dots are what the humans actually achieved. Orange dots are the result of applying the greedy algorithm to the ﬁnal state of human play, which is what the humans could have achieved without obstinacy. Vertical lines thus show the price of obstinacy. The dotted line indicates equality of the two scales. The open circles represent the average values over all scenarios.

There are two parts to this story, though, because solving these problems involves both selecting edges and closing deals on them. The greedy algorithm does not address the deal- closing issue and per force never leaves a potential deal unclosed; the humans often did. In 36 of the 54 experiments, the solution found by the human subjects was not even maximal – there were adjacent vertices that both could have closed another deal. Presumably this was because they simply could not agree on a split. However, the humans left the system in a state that could be improved post facto. We started the greedy algorithm in the ﬁnal state the humans reached and allowed it to attempt to ﬁnd more deals, thus producing a new state with no further unclosed deals. In all cases, this new state had a higher social

26 eﬃciency than the greedy algorithm achieved alone. This is shown in the orange dots of Figure 2.5. A line connects the human performance to the potential human performance, and we might dub this diﬀerence the price of obstinacy. In total, 7.9% of the money was “left on the table”, but 4.5% was due to obstinacy (more than half the lost value). We conclude that the humans found better matchings in the graph, and hence their behavioral price of anarchy is lower (better) than the greedy algorithm. But due to their additional obstinacy, their overall performance was no better.

2.5.2 Nodal Diﬀerences

There was much evidence that nodal income depends on its deal limit, its degree, and properties of the non-local neighborhood.

Unequal Splits

Most theoretical models (for example, the Yamagishi-Cook model) that apply game-theoretic rationality to bargaining suggest that deals in some networks will be split in an unequal fashion. We will report the splits using their inequality value (Ineq), defined as the absolute difference between the two fractional shares. It ranges from 0 (equal sharing) to 1 (one player gets everything). A total of 1271 deals were closed in all the 54 experiments, and 423 of them were split equally (inequality=0). But most were not split equally, every possible granular division was used for some splits, and 6 edges even had inequality=1 (which is surprising in itself since one partner gains nothing by signing the deal). The histogram in Figure 2.6 shows the inequality values. For comparison, we also show the histogram (in orange) from our preliminary session, which had no deal limits and produced an overwhelming portion of deals that split 50:50. The average inequality value over all games in our main session was 0.2097, which is a ratio of about 60:40, It thus seems clear that deal limits are invoking a significant increase in imbalanced splits.

27 40

20 occurrences

0 0 0.2 0.4 0.6 0.8 1. Inequality value

Figure 2.6: Histogram of deal splits in all games. The orange distribution was from our preliminary session, and the bar at 0 (equal shares) goes to 82%. The blue bars are from our main session, where we obtained a much greater spread of unequal splits.

Inequality and Eﬃciency

There is a significant correlation between average inequality value and the social efficiency achieved in each scenario — that is, when the subjects collectively tolerate greater inequality of splits, social welfare improves. These data are plotted in Fig. 2.7. The correlation coefficient is 0.52 with a confidence level of p = .027. Interestingly, in an earlier session of experiments without deal limits, the same correlation is highly negative.

Degree Distribution and Equality

How well does degree distribution predict wealth distribution? We examined the PL networks to answer this. The average inequality value in closed deals is 0.23 over the 3 PLPuniq games (where nodes tend to be adjacent to nodes of similar degree), while it is 0.36 for both PLNuniq (where nodes tend to be adjacent to nodes of very diﬀerent degree) and PL0uniq (where nodes tend to be adjacent to nodes of various degrees). The inequality values of the PLPuniq experiments are less than the joint PL0uniq and PLNuniq outcomes with a one-sided p<0.02. This indicates that nodes that have an opportunity to bargain with at least one other of similar degree have more power than one

28 0.35

0.30

0.25

coefficient 0.20

Gini 0.15

0.10

0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 social efficiency

Figure 2.7: Average inequality value versus Social Eﬃciency. The 9 PL networks are plotted in orange, and the 8 regular networks are plotted in blue. that is forced to bargain only with higher-degree nodes. (These are all unique-exchange games, so deal limit is not playing any distinguishing role.) These three networks all have the same degree distribution. Hence, degree distribution is not suﬃcient to predict inequality of wealth, even in unique-exchange networks.

Deal Limit Distribution and Equality

A similar story holds for deal limit distribution. Even if the distribution of deal limits for two networks are identical, the experimental results can differ widely based on whether a node bargains with nodes of similar deal limits or differing deal limits. In Torus-Uniform, all vertices have the same deal limit. In Torus-Rows, all vertices have two neighbors with the same deal limit (1 or 3) and two neighbors with a different deal limit (3 or 1). In Torus-Checker, all nodes are bargaining with nodes of a different deal limit. The average inequality values are 0.086, 0.13, and 0.36 for Torus-Uniform, Torus-Rows and Torus-Checker respectively. A means test shows these are all pairwise distinct with p<.03. These networks all have identical topologies. Thus, when network topology is not playing any distinguishing role, if vertices bargain with vertices of similar deal limits, the deals are on more equal terms compared to when vertices with differing deal limits bargain.

29 High Degree Confers Power

Over all closed deals in the PL graphs, the fractional take per closed deal of each node has a correlation of 0.47 with the degree of that node. If, to reduce the confound of differing deal limits, the study is confined to just the PL*uniq graphs, then the correlation coefficient is 0.59. Both these correlations are highly statistically significant. Thus, bargaining power increases with the size of the local market, at least in the setting where deal limits constrain behavior.

High Deal Limit Undermines Power

While higher degree confers bargaining power, higher deal limits had the opposite effect. The Torus-Rows graph was designed specifically for testing the effect of deal limit on a node’s bargaining power. In this graph, all nodes are identical up to relabeling, but half of them have deal limit 1 and half have deal limit 3. So if there is any systematic difference between two nodes’ bargaining power, the difference in their limits can be the only explanation. In the deals closed by limit-1 nodes, their mean fraction of the deal is 0.57. The limit-3 nodes obtained an average of 0.48. The difference was highly significant. (The two fractions do not add to unity because not all deals were between the two groups.) If only those deals between the two groups are considered, the fractions are 0.57 vs 0.43 and the difference is even more significant. The summary is that a higher deal limit confers less bargaining power.

Eﬀect of Global External Demand

The supply networks were designed to study the eﬀect of external demand. This property is not apparent at the node, but becomes apparent when contrasting the deal limits of two groups of nodes. In each supply network, the supply of deals from the left group (recall Figure 2.3) was manipulated to starve or overfeed the right group. How does the split of a deal depend on that relative supply? In the Undersupplied case the right nodes must compete among themselves for the attention of nodes in the left side, so we might expect their shares to be smaller than the

30 left side’s. In the Equisupplied case, the external demands are equal, so we might expect no diﬀerential in bargaining power. In the Oversupplied case, the left nodes must compete for deals from the right side, so we might expect their share of the deals to be smaller than the right.

external demand avg shares left right left right Undersupplied 30 39 0.57 0.43 Equisupplied 24 24 0.55 0.45 Oversupplied 20 14 0.52 0.48

Table 2.1: External demand imbalances shape bargaining results. The average splits shown are for edges between left and right nodes. Edges between right nodes have an average share of 0.5 by deﬁnition.

Table 2.1 shows the results. There is a correlation of -0.19 (p =0.01) between the external demand ratio and the deal share of the left nodes. The divisions favor the limit-2 nodes in all cases, consistent with the results of the previous section. However, that local property of relative limits is modulated by the global supply and demand ratio.

First Neighborhood Eﬀects

We examined the three PL*uniq scenarios to find effects attributable to the degrees of first (one-hop) neighbors. For both nodes in all deals in the PL*uniq games, compute the fraction of the node’s take and the average of the degrees of its neighbors. The correlation between these quantities is -0.60 and is highly significant. Similar results occur when the data are restricted to just those nodes with some fixed degree. The clear and consistent story in unique-exchange games is that the share obtained decreases as the average degree of the neighboring nodes increases. The opposite story holds when the first neighbors have higher deal limits. We compared the 4-regular networks torRows and torChkb. In torRows, a vertex of deal limit 1 has two neighbors of deal limit 1 and two neighbors of deal limit 3, while in torChkb, a vertex of deal limit 1 has all four neighbors with deal limit 3. The mean share of the former was 0.57

31 while the latter obtained 0.68. The diﬀerence is statistically signiﬁcant with p = .0001. The bargaining power of a vertex is enhanced when neighboring vertices have higher deal limits.

Second Neighborhood Eﬀects

How does the network effect propagate beyond the immediate neighborhood? The 2ndHood structure has two sets of 3 nodes, each of which have identical degree and first neighborhood degrees. The results of the previous section will be mute about how these nodes fare. However, the second (two-hop) neighborhood of these nodes are drastically different: the neighbors’ neighbors are leaves for 3 of them, and part of a clique for the other 3. The mean share of the first group was 0.347, the mean share of the second was 0.571, and the 2-sided p value was 0.027. The bargaining power of a vertex is enhanced if its neighbors’ neighbors have higher degree.

2.5.3 Comparison with Theoretical Models

We shall now point out some structural differences in solutions given by theoretical models and those found by human subjects. For our main session, where nodes have limits, we narrow our attention to the PL*uniq networks, since the Cook-Yamagishi model [31] was originally designed for unique exchange networks, and fails to make a stable prediction on the 2ndHood network. The model predicts that maximum social welfare (maximum matching) will be achieved on all the PL*uniq networks, which is rare in the experiments, as reported in Section 2.5.1. Further, the model predicts that a node with at least two leaves (nodes of degree 1) as neighbors always ends up with 1 fraction of a deal. This is due to myopic, rational − competition between the leaves, where is the smallest non-zero amount that can be received by a node by signing a deal (this is the granularity of offers available in the GUI, and we let =0.02). Accordingly, the model predicts that there should be at least 30 such skewed deals in our experiments with the PL*uniq networks. In contrast, we find that there is 1 deal where one node gets 100%, 5 deals where one node gets 98%, and only 10 deals where one node gets more than 90%. Further, all but one of these deals are between a leaf node and a node of degree 5 or 6. This indicates that extremely skewed deals are much rarer than

32 what game-theoretic rationale suggests, and is more likely when the degree differences are larger. In our preliminary session, where nodes have no limits, unequal splits are rare, as reported in Figure 2.6. I, in conjunction with my advisors Michael Kearns and Sanjeev Khanna, had designed a model for this setting – please see Chapter 3 for a detailed exposition. It predicts that all deals will be shared equally if the players at the nodes have linear utility functions, and network effects may arise only due to non-linearity of player’s utility. So the results of the experiments can be explained in this model if we assume that in our range of payoffs, the players have near-linear utility functions. This is not very surprising, since a player can make only a few dollars in each experiment.

2.5.4 Human Subject Diﬀerences

Humans were randomly assigned to nodes in each experiment and randomly reassigned in each replication of a scenario. Hence none of the results above could be ascribed to human diﬀerences. However behavioral literature is replete with examples of how human subjects leave their stamp, and some traits emerge in our data too.

Patience

The correlation between the average time for each human to close a deal and the average gain from closed deals, aggregated over all deals in all games, is 0.6664 (p =0.00). See data in Figure 2.8. Apparently, patience pays oﬀ.

Proposer vs. Acceptor

The user interface mechanism involved the following protocol for completing a deal: one player (proposer) makes an offer, the other player (acceptor) accepts that offer by matching it, and then the proposer closes the deal. This was not designed to provoke any asymmetry, but was intended to avoid unintended closed deals due to accidental mouse-clicks. Never- theless, by looking only at what the shares of the two parties were, can we say which party is more likely to have been the proposer? We find that indeed we can, and the party which gets the higher share is more likely to be the proposer. The mean proposer share across all

33 0.55 deal of

0.50 fraction

0.45 average

10 15 20 25 30 average time to closing seconds

Figure 2.8: People with patience win bigger splits. There are 36 dots here, corresponding to the 36 human players.

experiments was 53.6%, and the acceptor share was 46.4%. The Kolmogorov-Smirnov test 14 rejects the hypothesis that these shares come from identical distributions with p<10− . Some psychological eﬀect is clearly being expressed by this subtle asymmetry in protocol. All possible split ratios in closed deals were at least once proposed by someone (and accepted by someone), with the sole exception of 0:100%. The six cases where all the money went to one player were all proposed by the high-share side. It may be “irrational” for someone to agree to get 0, but it would have been even odder to see someone propose that he get 0.

The Eﬀect of Uncertainty in Costs

This last section is strictly about our preliminary experimental session, in which there were no deal limits imposed–so the limit on each node was effectively its degree. Here we found that social efficiency was higher when the players were simply uncertain about a particular detail regarding their neighbors. In the latter half of the preliminary session, we imposed varying transaction costs on nodes, which a node must pay for every deal it closes. The first half of the experiments had no costs. Occasionally during the latter half, we quietly imposed zero cost on every vertex.

34 This allows us to compare those games to the setting without costs. This cost was specific to each vertex. Only the player at that vertex, but not its bargaining partners, knew how much this cost was. We varied the costs significantly, from 0 up to 40% of the value of the deal. This generated enough uncertainty that in the few instances where every vertex had zero cost, no one could infer the costs of his partner. This 0-cost setting can be directly compared to the basic non-costed setting where every player knows that there are no costs involved. Hence the two situations were distinguished only by a lack of certainty. Players closed more deals in the (uncertain) 0-cost case than in the (known) no-cost case. The efficiency columns of table 2.5.4 show the fraction of possible deals that were closed in the two cases. The fraction went up in all 5 networks; the difference is significant with p = .004. Evidently, the level of obstinacy rises when people know for certain that their partner has no costs.

eﬃciency average std. dev. of inequality inequality non zero non zero non zero

PLP 0.85 0.96 0.012 0.01 0.033 0.037 PL0 0.84 0.93 0.009 0.009 0.025 0.029 PLN 0.72 0.93 0.027 0.012 0.057 0.051 cwc 0.84 0.95 0.015 0.023 0.035 0.076 2ndHood 0.84 0.97 0.014 0.009 0.040 0.044

Table 2.2: Social Eﬃciency and Inequality Values compared between no-cost case and zero- cost experiments. The cwc graph was a cycle with chords, shown in Appendix Figure 2.2. The others were as described for our main session.

Average inequality values of the deals and the standard deviations are also shown in the table. We expected the splits to be more uneven in the zero-cost case, but no consistent story was found.

35 2.6 Conclusion

The background theory is not yet prepared to describe all the phenomena we have observed here. Some bargaining theory suggests one party to a deal might get an infinitesimally small share, but our mechanism does not allow this. Hence our results cannot be exactly matched, but the scarcity of splits that are 98% or above seems to hint that the notion of “rationality” used by these theories needs to be adjusted. Other aspects of our results support theoretical models, notably the finding that phenomena at odd and even-length distances from a node alternately enhance and detract from the node’s earnings. The findings peculiar to people –namely the prevalence of obstinacy, the value of patience, the effect of protocol in the closing of a deal, and the state of knowledge about the partners– are all in need of theoretical development.

36 Chapter 3

Networked Bargaining with Non-Linear Utilities

In this chapter, we study a networked bargaining game through theoretical modeling of rational agents, which leads us to an equilibrium concept. Results in this chapter were published in collaboration with Michael Kearns and Sanjeev Khanna [26]. We shall refer to the network as a bargaining network, each edge or transaction as a deal, the participants in the transaction as endpoints of the deal, and an agreement between its endpoint nodes to share the surplus as signing of the deal. We shall also say that a deal is incident on its two endpoints. An outcome or state in a bargaining network is a specification of how each deal gets shared, i.e. the amount received from a deal by each of its two endpoints. Networked bargaining game has been studied heavily in network economic theory and networked social exchange theory. Most of these models place explicit constraints, such as a limit on the number of deals that a player can sign, to induce competition among players. In the absence of such constraints, virtually all of these solution concepts show zero effect of network structure – each edge is shared independently. At the same time, all these models in the literature, that have a game-theoretic formulation, either implicitly or explicitly assume that players have linear and often identical utility functions. Our primary goal in this chapter is to understand whether and how non-linear utility among players may affect the outcome, and in particular, whether it can introduce network effect. So to isolate the

37 network effect caused by non-linear utility, we choose not to place any additional constraint that will have their own network effect. Further, we shall place identical utility functions on all the players. In the presence of other constraints, network effects due to diminishing marginal utility will get superimposed with network effects due to the constraints. As we shall see, non-linear utility among players can introduce its own network effects. Throughout this chapter, we shall focus on the situation where players have diminishing marginal utility, that is, the utility function is concave. The nature of this effect can best be summarized as ”rich-get-richer”, that is, it enhances inequality of wealth. If bargaining power of a node is measured as the average share it receives from the deals incident on itself, then we find that the node’s bargaining power is usual enhanced if it has high degree and its neighbors have low degree. The precise effects are, of course, much more complex. In the context of networked social exchange, a network where players have concave utility may be viewed as a negatively connected network. Before describing our results in detail, it is necessary to discuss the solution concept we use in this chapter.

3.1 Overview of the Model

In our model, we assume that the amount of the surplus of an edge is known to both its endpoints. We consider a theoretical solution concept that proposes an outcome given a bargaining network. Naively speaking, the outcome can be viewed as a prediction of what might happen if human players played the game. Since it is unrealistic to expect actual human players to follow a precise theoretical model of rationality, it should instead be viewed as a fair outcome. Our theoretical solution concept naturally extends any two-player bargaining solution concept to a corresponding equilibrium concept over the networked game. Before proceeding to describe our model in detail, let us review the two-player concepts that will be necessary to understand our model.

Two-Player Bargaining Solution Concepts Bargaining has been studied extensively in economics and sociology, both theoretically and experimentally. A simple setting that appears frequently in the literature is when there are only two parties negotiating a single

38 deal. The deal yields a fixed total wealth if the two parties can agree on how to share or divide it; otherwise both parties receive nothing. There may or may not be external payments (outside options or alternatives) that parties receive, possibly depending on whether the deal gets signed. Bargaining solution concepts provide predictions about how the wealth will be shared, or what division is “fair”, which may depend on the player utility functions. There are several bargaining solution concepts for this two-player setting in economic theory, and here we shall focus on two of them: Nash Bargaining Solution (NBS) and Proportional Bargaining Solution (PBS). Both these solution concepts (and most others) predict that the division of wealth is a function of the additional utility (compared to some outside option or alternative) each player receives by accepting the deal, which we shall refer to as difference utility. NBS states that the division of wealth should maximize the product of the difference utilities of the two players, while PBS states that the division should maximize their minimum. When the players have increasing and continuous (in accrued wealth) utility functions, PBS simply states that the two players should have equal difference utility from the deal. We choose to focus on these two concepts because they are representatives of two broad classes of these solution concepts: NBS represents those solutions that do not allow direct comparison of utility across players, and are thus impervious to scaling of utility functions; whereas PBS represents those that permit such comparisons. For a comprehensive discussion of different bargaining solution concepts, please see [15].

Network Equilibrium Models We study a natural generalization of these two-player solution concepts to our multi-party, networked setting. We propose that an outcome of a bargaining network is an equilibrium if every edge is locally stable. An edge is said to be locally stable if it satisfies a chosen two-player bargaining solution concept. An outcome that satisfies the NBS (respectively, PBS) local stability on every edge is called an NBS equilibrium (respectively, PBS equilibrium). To completely specify our model, we need to define the notion of local stability. The key to defining local stability, as a parallel to two-player solutions, is to define the difference utility that a player receives by signing a particular deal. When a player has non-linear utility, difference utility from a deal in the network setting should depend not only on the share received on that deal, but also on the amount that a player receives

39 from other deals incident on it. Intuitively, if a player with diminishing marginal utility (in other words, a concave utility function) has earned a lot of money from the other deal, the additional utility he receives by signing this deal is going to be small. One way to view difference utility from a deal is the immediate loss in utility if the player does not sign that deal. Thus, if a player with utility function earns a sum of w from all deals incident on it, U and earns x from a particular edge, then the difference utility that the player receives from that edge is (w) (w x). Now, NBS proposes maximization of the product, while PBS U − U − proposes maximization of the minimum, of difference utilities of the endpoints of a deal, as local stability concepts. Our equilibrium concept can be viewed as the fixed points of the following selfish and myopic dynamics among the players: At each time-step, one (or more) player wakes up, chooses an edge incident on itself, and evaluates whether it is being unfairly treated on that deal, given the chosen two-player concept, the utility functions of the other endpoint and itself, and the wealth that the other endpoint and itself are receiving from all other deals. An unfair treatment corresponds to receiving a share that is smaller than what is stipulated by the two-player concept. If the player is indeed being unfairly treated, it wakes up the other endpoint, and renegotiates the deal to make it fair. This dynamics is also a generic algorithm for computing an equilibrium on a network. Our equilibrium concept is very similar to a model of balanced outcomes proposed by Cook and Yamagishi [31] and later analyzed by Kleinberg and Tardos [68], with two differences:

Cook and Yamagishi [31] imposes a limit on the number of deals that a player can • sign, which is often less than its degree (i.e. number of deals incident on the node). This limit introduced direct competition among nodes to catch the attention of their neighbors. They speciﬁcally considered unit limit on each node, but the model generalizes to arbitrary limits. Our setting can be viewed as a special case of [31] where limits are absent, or equivalently, the limit on each node equals or exceeds its degree.

Cook and Yamagishi [31] implicitly assumed that players have linear and identical • utility functions only, and so their solution concept has zero network eﬀect in the absence of limits. A consequence of identical linear utility function is that there is

40 no diﬀerence between outcomes proposed by Nash and Proportional bargaining. We consider non-linear utility, in particular concave utility, and show that network eﬀect occurs even in the absence of limits. We elaborate on this point below in Section 3.1.1.

3.1.1 Overview of Results and Techniques

The first and easily observable point is that if players have linear utility functions, then network structure has no effect on the outcome – each deal is shared independently, possibly depending only on the utility functions of the endpoints. This is simply because wealth accrued from other edges have no effect on difference utility. A formal characterization of all PBS and NBS equilibria, when players have linear utility functions, is provided in Section 3.3. The main contribution of this chapter is the following observation: network structure plays an effect simply due to diminishing marginal utility of players. This is true for both PBS and NBS equilibria. To begin, we establish existence of equilibrium using a fixed point argument in Section 3.4. The structural observation for a two-player setting that forms the core reason for the network effects on equilibrium is the bargaining monotonicity condition, described in Section 3.5. This condition states that on any edge (u, v), if the outside options of u increases while that of v decreases, then u claims a higher share of wealth on this edge when it is renegotiated. It is satisfied by all concave utilities when we consider a PBS equilibrium, and it is also satisfied by natural smooth concave functions such as xp when we consider a PBS equilibrium. This monotonicity condition is the reason behind the “rich-get-richer” network effects that we observe in our model. Consider a star network, where multiple leaves attached to a center of a star, the center can negotiate some share from all the leaves sequentially, and then renegotiate all of them for larger shares. This is because a gain in renegotiating one deal translates to a gain on all other deals when renegotiated, if strict monotonicity is satisfied. So the center of the star gets the large share on all edges. Dynamics of this kind is seen in larger networks as well. We next present a fully polynomial time approximation scheme (FPTAS) to compute approximate network bargaining equilibria in bipartite networks when the utility functions

41 and the solution concept satisfy the bargaining monotonicity condition. Bipartite networks are natural in many settings in which there are two distinct“types” of players — for instance, buyers and sellers of a good. The bargain monotonicity condition is satisfied by all concave utility functions in the PBS concept, and by natural utility function classes such as xp, 0 < p<1 and log(a + bx),a>0,b > 0 in the NBS concept. The algorithm can be viewed as iterating best-response dynamics on each edge under a particular schedule of updates; we show that for this schedule, the algorithm converges to (additive) -approximate equilibrium 1 in time polynomial in − and the size of the input, if the values of wealth on the edges are polynomially bounded (for multiplicative approximation, we can handle arbitrary edge values). Whether the particular schedule for which we can prove fast convergence can be generalized is an interesting open problem. We also show that NBS equilibrium has less unequal sharing than PBS equilibrium, on an edge where both endpoints have high degree. This result is proved in Section 3.7. We also perform simulation and statistical analyses of the effects of network structure on the wealth distribution at equilibrium for the two solution concepts, on networks randomly chosen from well-studied formation models (such as preferential attachment and Erdos- Renyi), and for a range of utility functions. Empirically we find that wealth of a vertex is highly correlated with degree, but degree alone doesn’t determine wealth. We also find that bargaining power of a vertex, measured as the average share received by the vertex on all its edges, appears to increase with degree. Finally, we find stark differences between wealth distribution in PBS and NBS equilibrium. We find that the network effect is more pronounced in PBS equilibrium than in NBS equilibrium, which is manifested in two ways: first, the variation of bargaining power is larger in a PBS equilibrium than in an NBS equilibrium; and second, a higher number of edges have a highly skewed split in PBS than in NBS equilibrium. We also observe how network effects decrease as the utility functions approach linearity. Finally, we show that neither PBS nor NBS equilibrium is unique on bargaining networks in general. Uniqueness is an important and preferred property, since it can serve as a prediction of how the wealth will be divided, as well as a measure of fair division. Un- fortunately, we find non-uniqueness for the class of regular graphs with unit wealth on all

42 edges, and the same utility function for all vertices. This class is interesting because every network in this class has one state which is both an NBS as well as a PBS equilibrium: the state where every edge is divided fifty-fifty. This state is also superficially fair, given the symmetry of the opportunities that the players have, as well as their behavior. Further, this state is the unique equilibrium when utility is linear, so it is natural to ask if the network has any “unfair” equilibrium for concave utility functions. On a positive note, we recognize conditions on utility that makes PBS and NBS equilibrium in this class unique, and U show that natural concave functions such as (x)=xp for any 0

3.1.2 Related Work

The network setting that we consider falls under the heavily studied field of network exchange theory, which was pursued primarily by sociologists in the context of bilateral social exchanges. Most models consider that every player has a limit on the number of deals it can get into, which is less than or equal to its degree in the network – that is, there is no imposed limit other than the degree itself. Note that our setting corresponds exactly to this limited exchange setting when the limit for every player is equal to or exceeds her degree. Several models have been proposed to predict what agreements the players will get into, and how will the wealth on these deals get shared (eg. [31, 74, 51, 14, 17]). Skvoretz and Willer [83] conducted human subject experiments to practically verify these theoretical predictions. Most of the focus has, in fact, been on unique exchanges, where every player can get into only one deal. Recently, Kleinberg and Tardos [68] analyzed the model given by Cook and Yamagishi [31] in the unique exchange setting, and found an elegant theoretical characterization, connecting the bargaining solutions to the theory of graph matchings. All these models assume that all the players have the same behavior, and focus on the differences in bargaining power caused by network structure only. They also agree that these differences arise from the threat of exclusion, that is, a vertex can get into only a few deals and so has to ignore the offers of some of its neighbors; so it can ask for a better offer from its neighbor by threatening to get into a deal with some other neighbor instead. In absence of limits, there is no threat of exclusion, and all the models predict that there will

43 be no network effect, and the wealth is divided into two equal shares on every edge. However, it is important to note here that all these models implicitly or explicitly assume linear utility functions. Our model takes into account that players may have non-linear utility. In particular, we focus on increasing concave utility functions, that is, those with diminishing marginal utility. Diminishing marginal utility is a well-known phenomenon, and is also often used in financial theory to capture risk aversion. Our model agrees with these previous models when the utility functions are linear. In fact, when the utility functions are linear, the concept of PBS equilibrium is identical to the equi-dependence theory of Cook and Yamagishi [31]. However, interesting network effects appear when the players have concave utility functions, even if all players have the same utility function. As mentioned before, a closely related literature is the study of sequential offer bargaining in networked setting [34, 27, 73, 1, 2], where the surplus shrinks with the passage of time, and the negotiation proceeds through controlled rounds of alternating offers and counter-offers. This is in contrast to our experiments, where either player can make an offer at any time on a deal, and the surplus remains unchanged over time, though there is a deadline at which the game ends. The equilibrium concept studied here is subgame perfect equilibria in a game where players have complete information about the entire network. This is the major distinction of this line of work from our local equilibrium model, in which players take myopic decisions solely based on local dynamic information. This solution concept also suffers from drawbacks as a predictive model similar to those outlined above, such as multiplicity of equilibria. Another group of concepts related to our setting arise in coalitional games, such as Shapley value, core, kernel, nucleolus and bargaining sets. These concepts often involve the ability of players to form arbitrary coalitions, which implicitly assumes that the players have information about the entire network, and are not acting myopically based on local information. Thus, these concepts assume that players can solve problems that are computationally difficult. In fact, Deng and Papadimitriou [41] showed that many of these concepts are computationally hard, and suggested that a solution concept is appropriate only if it is efficiently computable. In strong contrast, our model only expects simple selfish behavior from the players. Perhaps not unrelated to this aspect of our model, we shall show

44 that equilibria in our concept is computable in polynomial time on bipartite graphs, and natural heuristics perform well in our simulations on random graphs.

3.2 Preliminaries

A bargaining network is an undirected graph G(V,E)withasetV of n vertices and a set E of m edges, where vertices represent player and edges represent possible bilateral deals. There is a positive value c(e) associated with each edge e E, which is the wealth on that ∈ edge. There is also a utility function for each player v. The utility functions are all Uv represented succinctly and are computable in polynomial time.

Let e1,e2 ...em be an arbitrary ordering of the edges in E,whereei has endpoints ui and v , i =1, 2 ...m.Astate of the bargaining network is described by the division of i ∀ wealth on each edge of the graph. Let x(ui,ei) and x(vi,ei) denote the wealth ui and vi receive from the agreement on the edge ei, respectively. Since we are interested in the structure of equilibrium, so without loss of generality, we shall always assume that x(v ,e )=c(e ) x(u ,e ), that is the wealth on the deal has been realized. We shall i i i − i i m represent a state of the bargaining network as a vector s =(s1,s2 ...sm) R such that ∈ si = x(ui,ei). Note that s uniquely determines the division of wealth on all edges. Let s Rm be a state of the bargaining network G. For any vertex u and any edge ∈ e incident on u,letγs(u) denote the total wealth of a vertex u from all its deals with its neighbors. Let xs(u, e) denote the wealth u gets from the agreement on edge e. Let α (u, e)=γ (u) x (u, e) be the wealth u receives from all its deals except that on e.We s s − s say that αs(u, e) and αs(v, e) are the outside options with respect to the edge e for u and v respectively, that is, the amount each of them receives if no agreement is reached on the deal on e.

Definition 3.2.1 (Differential Utility). Let s be any state of the bargaining network. Let x be the wealth of u from the deal on e =(u, v). Then, the difference utility of u from this deal is ∆ (x)= (α (u, e)+x) (α (u, e)),andthedifference utility of v from this deal u Uu s − Uu s is ∆ (c(e) x)= (α (v, e)+c(e) x) (α (v, e)). v − Uv s − − Uv s We shall drop the subscript s if the state is clear from the context.

45 Definition 3.2.2. Let s be any state of the bargaining network. Define ys(u, e) to be the wealth u would get on the edge e =(u, v) if it is renegotiated (according to some two- party solution), the wealth divisions on all other edges remaining unchanged. Also define change(s, e)= x (u, e) y (u, e) . | s − s |

3.2.1 Proportional Bargaining Solution (PBS)

We say that the allocation on an edge e =(u, v) with value c satisﬁes the Proportional Bar- gaining Solution (PBS) condition if it maximizes the function W (x)=min ∆ (x), ∆ (c(e) P { u v − x) where x denotes the allocation to u. Thus if an edge e is renegotiated according } to the Proportional Bargaining Solution (PBS), then ys(u, e) = arg max0 x c WP (x) and ≤ ≤ y (v, e)=c(e) y (u, e). Note that y (u, e) is simply a function of the two values α (u, e) s − s s s and αs(v, e), along with the utility functions of u and v. The following lemma gives a simpler equivalent condition for PBS when the utility functions are increasing and continuous, and is applicable to the two-party setting as well.

Lemma 3.2.1. If the utility functions of all vertices are increasing and continuous, then for any edge e =(u, v), the PBS condition reduces to the condition ∆ (x)=∆ (c(e) x), u v − that is, the condition of equal diﬀerence utility, and there is a unique solution x satisfying this condition.

Proof. Since the utility functions are increasing, so ∆u(x) is an increasing function while ∆ (c(e) x) is a decreasing function of x, and both are non-negative functions when x v − ∈ [0,c(e)]. Also, ∆u(0) = ∆v(0) = 0. Thus there is a unique value z such that ∆u(z)= ∆ (c z). If xz,then∆ (x) > ∆ (c(e) x), and so W (x)=∆ (c(e) x) < ∆ (c(e) z). So W (x) u v − P v − v − P takes its maximum value only at x = z.

3.2.2 Nash Bargaining Solution (NBS)

We say that the allocation on an edge e =(u, v) satisﬁes the Nash Bargaining Solution (PBS) condition if it maximizes the function W (x)=∆ (x)∆ (c(e) x)wherex denotes N u v − the allocation to u. Thus if an edge e is renegotiated according to PBS, then ys(u, e)=

46 arg max0 x c WN (x) and ys(v, e)=c(e) ys(u, e). Note that ys(u, e) is simply a function ≤ ≤ − of the two values αs(u, e) and αs(v, e), along with the utility functions of u and v.

If e is renegotiated according to the Nash Bargaining Solution (NBS), then ys(u, e)is a value 0 x c such that the NBS condition is satisﬁed, that is, the function W (x)= ≤ ≤ N ∆ (x)∆ (c(e) x) is maximized. u v − The following lemma gives a simpler equivalent condition for NBS when the utility functions are increasing, concave and twice diﬀerentiable, and is applicable to the two-party setting as well.

Lemma 3.2.2. If the utility functions of all vertices are increasing, concave and twice differentiable, then for any edge e =(u, v), the NBS condition reduces to the condition ∆u(x) = ∆u (x) ∆v(c(e) x) ∆ (c(e)−x) , that is, the condition of equal diﬀerence utility, and there is a unique solution x − v − ∆u(x) ∆v(c(e) x) satisfying this condition. Moreover, let Qu(x)= ∆ (x) , and let Rv(x)= ∆ (c(e)−x) . Then u − v − Q (x) is increasing, R (x) is decreasing, and Q (x) R (x) has a unique zero in [0,c(e)]. u v u − v Proof. Diﬀerentiating ∆ (x)∆ (c(e) x)withrespecttox and equating the derivative to u v − zero, we get the equation

∆ (x)∆ (c(e) x)+∆ (x)∆ (c(e) x)=0 u v − u v − ∆ (x) ∆ (c(e) x) u = v − ⇒ ∆ (x) −∆ (c(e) x) u v − Since the utility functions are concave and increasing, so is ∆ (x), while ∆ (c(e) x) u v − is concave and decreasing in x, and both functions are positive. Thus we have ∆ (x) u ≥ 0, ∆ (c(e) x) 0, ∆ (x) > 0, ∆ (c(e) x) < 0, ∆ (x) < 0, and ∆ (c x) < 0. v − ≥ u v − u v − Diﬀerentiating ∆ (x)∆ (c(e) x) twice and using this information, we can easily see that u v − this second derivative is always negative. Thus a zero of the ﬁrst derivative is a maxima. Thus we have proved the equivalence of the two conditions. ∆ (x) ∆ (c(e) x) To show uniqueness, let Q (x)= u , and let R (x)= v − . Again, we can u ∆u(x) v ∆v(c(e) x) − − verify that Q (x) > 0 and R (x) < 0 since ∆ (x) < 0 and ∆ (c x) < 0. Moreover, u v u v − Qu(0) = 0 and Rv(c) = 0, since ∆u(0) = ∆v(0) = 0. Thus, by an argument similar to the proof of Lemma 3.2.1, we know that the increasing function Qu(x) and the decreasing function Rv(x) become equal at a unique value of x.

47 3.2.3 Stability and Equilibrium

Deﬁnition 3.2.3 (Exact Stability and Equilibirum). We say that an edge e is stable in a state s if renegotiating e does not change the division of wealth on e,thatis,change(s, e)= 0.Wesaythatastates is an equilibrium if all edges are stable.

We also study two notions of approximation, namely, additive and multiplicative approximations, as deﬁned below.

Deﬁnition 3.2.4 (Additive -Stability and Equilibrium). We say that an edge e is -stable in the additive sense in a state s if change(s, e) < .Wesaythats is an additive - approximate equilibrium if all edges are additive -stable.

Deﬁnition 3.2.5. (Multiplicative -Stability and Equilibrium) We say that an edge e is -stable in the multiplicative sense in a state s if y (u, e) x (u, e) < x (u, e) and | s − s | s y (v, e) x (v, e) < x (v, e).Wesaythats is a multiplicative -approximate equilibrium | s − s | s if all edges are multiplicative -stable.

In this chapter , an approximate equilibrium will refer to additive approximation, unless speciﬁed otherwise. We refer to an equilibrium as an NBS equilibrium if the renegotiations satisfy the NBS condition. We refer to the equilibrium as a PBS equilibrium if the renegotiations satisfy the PBS condition.

3.2.4 Bargaining Concepts as Nash Equilibria

The bargaining solutions may also be viewed as pure Nash equilibria of certain games. Each edge is a player, and an edge e has a strategy space [0,c(e)]. Strategy of an edge corresponds to the division of wealth on it. Let e =(u, v) be an edge playing strategy x.Ifthepayoff of e is ∆ (x)∆ (c(e) x), and each edge wishes to maximize its own payoff,thenthepure u v − Nash equilibria of this game are exactly the NBS equilibria of the network. Similarly, if the payoff is instead defined to be min ∆ (x), ∆ (c(e) x) , then the pure Nash equilibria of { u v − } the game coincides with PBS equilibria. Thus, updating an edge corresponds to an edge playing a best response move in the corresponding game. As with all pure Nash equilibria concepts, we thus have a natural

48 heuristic that gives an equilibrium if it terminates: start from an arbitrary state, and then update unstable edges repeatedly till all edges are stable. This heuristic is called best response dynamics. It is worth noting that approximate equilibria of this game does not necessarily coincide with approximate equilibria of the bargaining network with the same approximation factor.

3.3 Linear Utility Functions: Characterizing All Equilibria

In this section, we characterize all possible NBS and PBS equilibria when all vertices have linear increasing utility functions, for every vertex v. We show that in our model, if we make this assumption, there is a unique NBS equilibrium and a unique PBS equilibrium, and network topology has no inﬂuence on the division of proﬁts on the deals at equilibrium. The following two theorems formalise these observations.

Theorem 1. Suppose all vertices have linear increasing utility functions. Then there is a unique NBS equilibrium, in which the proﬁt on every edge is divided equally between its two end-points.

Proof. Let (x)=k x + l i V (G). Let s be an NBS equilibrium. Then, on any edge Ui i i ∀ ∈ e =(u, v), a (x)=k x and b (x)=k (c(e) x). In particular, they are independent of s u s v − αs(u, e) and αs(v, e). Since ku and kv are constants, the product as(x)bs(x) is maximized when x = c(e)/2.

Theorem 2. Suppose all vertices have linear increasing utility functions. Let (x)= Ui k x + l i V (G). Then there is a unique PBS equilibrium, such that for any edge i i ∀ ∈ e =(u, v), x (u, e)=c(e) kv . s ku+kv

Proof. Since the utility functions are increasing and continuous, we can apply Lemma 3.2.1 and for any edge e =(u, v), solve the equation as(x)=bs(x). Substituting the values, we

kv get the equation kux = kv(c(e) x), which has the unique solution x = c(e) . Again, − ku+kv the solution is independent of αs(u, e) and αs(v, e). Further, if ku = kv, then the solution becomes x = c(e)/2, which is the same as the solution for NBS.

49 3.4 Existence of Equilibrium for Non-Linear Utility Func- tions

We now turn our focus towards non-linear utility functions. In this section, we prove that PBS and NBS equilibria exist on all graphs, when the utility functions satisfy some natural conditions. The proofs use the Brouwer ﬁxed point theorem, and is similar to the proof of existence of mixed Nash equilibrium in normal form games.

Theorem 3. PBS equilibrium exists on any social network when all utility functions are increasing and continuous. NBS equilibrium exists on any social network when all utility functions are increasing, concave and twice diﬀerentiable.

Essentially, it is suﬃcient for the utility functions to satisfy the following general condition of continuity:

Condition 1. Let s be any state of the bargaining model, and e =(u, v) be an edge. For every > 0, there exists δ > 0 such that for any state t such that α (u, e) α (u, e) < δ | t − s | and α (v, e) α (v, e) < δ, we have y (u, e) y (u, e) < . | t − s | | t − s |

Note that ys(u, e) and yt(u, e) are inﬂuenced both by the utility functions as well as the two-party solution concept that is used (NBS or PBS). Thus whether Condition 1 holds will depend on whether the renegotiations follow the NBS or the PBS condition, and also on the utility functions.

Lemma 3.4.1. If Condition 1 holds for the NBS solution concept or the PBS solution concept, then NBS or PBS equilibrium exists, respectively.

Proof. We define a function f :[0, 1]m [0, 1]m that maps every state s to another state → f(s). Given s, we can construct the unique solution t such that the deal on an edge e =(u, v) in t is the renegotiated deal of e in s, that is, xt(u, e)=ys(u, e). We define f(s)tobet. Thus, f(s) is the “best-response” vector for s. Clearly, s is an -approximate equilibrium if and only if s f(s) < . In particular, || − ||∞ s is an equilibrium if and only if f(s)=s, that is, s is a fixed point of f. Also, [0, 1]m is a closed, bounded and convex set. So if f were continuous, then we can immediately

50 use Brouwer ﬁxed point theorem to deduce that the equilibrium exists. Thus the following claim completes the proof.

Claim 1. f is continuous if and only if Condition 1 holds.

Proof. Suppose Condition 1 holds for some and δ.Thus,if s t < δ/n,then αt(u, e) || − ||∞ | − α (u, e) < δ and α (v, e) α (v, e) < δ, so, by Condition 1, y (u, e) y (u, e) < , and s | | t − s | | t − s | thus f(s) f(t) < .Sincethereexistsaδ for every > 0, so f is continuous. || − ||∞ Now suppose f is continuous. Let > 0. Then there exists δ > 0 such that for any solution t,if s t < δ,then f(s) f(t) < , which implies that coordinatewise for || − ||∞ || − ||∞ every edge e,wehave y (u, e) y (u, e) < . Since this is true for all > 0, Condition 1 | t − s | holds.

Lemma 3.4.2. Condition 1 holds for all increasing, continuous utility functions when renegotiations follow the PBS condition.

Proof. Let s be any state of the bargaining model and let e =(u, v) be any edge. Here, Lemma 3.2.1 is applicable. Let h(s, x)=a (x) b (x). Also, let g (x)=h(s, x)bea s − s s function deﬁned on a particular state s. Note that gs is an increasing, continuous function on the domain [0,c(e)], gs(0) < 0 and gs(c(e)) > 0. The renegotiated value ys(u, e)isthe unique zero of gs(x) between 0 and c(e). Let y = y (u, e) be the zero of g . Let η = max g (y ) , g (y + ) .Then,sinceg s s {| s − | | s |} s is increasing, η > 0, and for all x [0,c(e)] (y ,y+ ), g (x) η. ∈ \ − | s | ≥ Now, observe that h(s, x) is dependent on αs(u, e), αs(v, e) and x only, and is continuous in all three of them when the utility functions are continuous. Thus, there exists δ > 0 such that for any state t where α (u, e) α (u, e) < δ and α (v, e) α (v, e) < δ,wehave | s − t | | s − t | h(s, x) h(t, x) < η x [0,c(e)], that is g (x) g (x) < η. This implies that g (x) =0 | − | ∀ ∈ | t − s | t for all x [0,c(e)] (y ,y+ ), and so the zero of g ,whichisy (u, e), lies in the range ∈ \ − t t (y ,y+ ). − Lemma 3.4.3. Condition 1 holds for all increasing, concave and twice diﬀerentiable utility functions when renegotiations follow the NBS condition.

51 Proof. Let s be any state of the bargaining model and let e =(u, v) be any edge. Here, Lemma 3.2.2 is applicable. Let h(s, x)=q (x) r (x). Also, let g (x)=h(s, x). The rest s − s s of the proof identically follows that of Lemma 3.4.2.

Combining Lemmas 3.4.1, 3.4.2 and 3.4.3, we get Theorem 3.

3.5 The Bargain Monotonicity Condition

The bargain monotonicity condition seems like a natural condition that a negotiation between two selfish players can be expected to satisfy. This condition is indeed satisfied by the PBS solution concept whenever the utility functions are concave and increasing. For NBS, however, concavity and monotonicity of utility functions is not sufficient for satisfying this condition. We will instead identify a stronger property of utility functions that is necessary and sufficient for satisfying the bargain monotonicity. This stronger property is satisfied by several natural classes of utility functions including xp, 0 0.

Condition 2 (Bargain Monotonicity Condition). An instance of the bargaining problem satisﬁes the bargain monotonicity condition with respect to a solution concept (PBS, NBS) if for any edge e =(u, v) and a pair of states s and s of the bargaining network such that e is stable in both s and s with respect to the solution concept, whenever α (u, e) α (u, e) s ≥ s and α (v, e) α (v, e), then x (u, e) x (u, e). s ≤ s s ≥ s The above condition states that on any edge (u, v), if the outside options of u increases while that of v decreases, then u claims a higher share of wealth on this edge when it is renegotiated. Note that Condition 2 is essentially a two-player condition that has no dependence on the network itself and merely depends on the outside options of the players. The bargain monotonicity condition seems like a natural condition that a negotiation between two selﬁsh players can be expected to satisfy. It is a condition that depends on the bargaining network as well as the solution concept.

Lemma 3.5.1. PBS solutions satisfy the bargaining monotonicity condition on all networks where the utility functions of all the players are concave and increasing.

52 Proof. Let p = xs(u, e) and q = xs(v, e). Consider the state s derived from s with the sole modification that u gets p and v gets q on e.Then,thedifference utility of u from e in s is at most that in s , since the function (z + p) (z) is decreasing in z when is concave U − U U (this is precisely equivalent to diminishing marginal utility). By a symmetric argument, the difference utility of v from e in s is at least that in s.Thus,thedifference utility of e to u is at most that of e to v in s. So by Lemma 3.2.1, u must get at least p on the edge e in s to ensure that e is stable.

We now focus on identifying utility functions where the NBS concept will satisfy Con- dition 2 on all networks. Let be the utility function of any vertex u. In light of Lemma U (α+x) (α) 3.2.2, it is clear that R(α,x)= U (α+−xU) must be a non-increasing function of α.If U not, then there exists some positive α1 and α2 > α1, such that R(α1,x)

Lemma 3.5.2. Let χ be the family of all concave, increasing, and twice diﬀerentiable utility (α+x) (α) functions χ, such that R(α,x)= U (α+−xU) is a non-increasing function of α for all U ∈ U α > 0 and x>0. Then every network, where all players have utility functions from χ, satisﬁes the bargain monotonicity condition for the NBS concept.

Now, suppose that is concave, increasing and twice diﬀerentiable at all positive values. U Simplifying the equation, we have

d (α + x) (α) U − U 0 dα (α + x) ≤ U (α + x) (α) (α + x) (α) U − U U − U ⇔ (α + x) ≤ (α + x) U U (since (α + x) > 0, and (α + x) 0) U U ≤ It can be easily veriﬁed that natural utility functions such as xp, 0 0,b>0,f 0 belong to χ, and so Lemma 3.5.2 applies. ≥

53 Not all concave utility functions satisfy the equation above. In particular, we construct a concave, increasing and twice differentiable function that violates the equation. The key is U to construct a sharp change in marginal utility. We achieve this by making (α + x) very |U | large compared to the other expressions in the equation. We describe the utility function by defining . Let (x) = 1 for 0 1.01. For 1 x 1.01, U U U ≤ ≤ (x) decreases from 1 to 1/2 smoothly, so that is differentiable everywhere, and we can U U also ensure that (1.005) < 50, which is the average slope of in the range [1, 1.01]. U − U Now, in the above equation, let α =0.5 and x =0.505. Note that the left-hand-side

x (α+x) of the equation is more than U (α+x) = x>0.5, while the right hand side is less than U 1/ (α + x) < 1/50, thus violating the equation. So we conclude that if the marginal |U | utility of a player changes abruptly, Condition 2 may be violated. The above discussion is summarized in the following lemma.

Lemma 3.5.3. There exists a bargaining network where all the players have concave utility functions, such that Condition 2 is not satisﬁed in the NBS concept.

In contrast, however, natural utility functions such as xp, 0 0,b > 0,f 0 satisfy the condition that R(α,x) is a non-increasing function of ≥ α, and so Lemma 3.5.2 applies. Thus Condition 2 is satisﬁed when all players have utility functions of these forms. The key reason is that there is no sharp change in marginal utility in these functions. To illustrate, we show our computation for (x)=xp, 0 p<1, below. U ≤ Let (x)=xp for any p between 0 and 1. We show that R(α,x) is a non-increasing U function of α.

d (α + x) (α) d (α + x)p αp U − U = − dα (α + x) dα p(α + x)p 1 U − p 1 p 1 p p d α + x α (α + x) 1 α + x − α = − = 1 p (1 p) dα p − p p − α − − α + x 1 1 p p α + x = (1 pz − (1 p)z− )wherez = p − − − α To prove that the above derivative is non-positive, it is suﬃcient to show that g(z)= pz1 p +(1 p)z p 1 z 1. This follows from observing that g(1) = 1, and that − − − ≥ ∀ ≥

p (1+p) p 1 g(z)=p(1 p)z− p(1 p)z− = p(1 p)z− (1 ) 0 when z 1 − − − − − z ≥ ≥ 54 3.6 Computing Equilibria

We will now design a fully-polynomial time approximate scheme for computing approximate additive and multiplicative equilibria in bipartite networks, provided the bargain monotonicity condition below is satisﬁed.

3.6.1 Algorithmic Results

To be used as a subroutine in our algorithm, we deﬁne an Update oracle, that takes an edge e =(u, v) of the network as input. The oracle is called when our algorithm shall have a current state s of the bargaining network, and the oracle shall renegotiate the edge e according to the 2-player bargaining solution we use, and modify the division of wealth on e only, to change the state to s, so that e becomes stable. The computation of the Update oracle depends solely on the outside options of u and v with respect to e, that is, αs(u, e) and αs(v, e), and the utility functions of the u and v. We assume that the oracle also knows the input to the problem, that is, the social network and the utility functions of the players, as well as whether the goal is to compute an NBS or a PBS equilibrium. The Update oracle essentially performs an improvement step of the best response dynamics in the game played by edges that was described in Section 3.2.4. Since the bargaining equilibrium concepts are pure Nash equilibria of this game, it is natural to wonder if a sequence of updates starting from a random state of the bargaining network converges to equilibrium. We do not know if all sequences converge, though our simulations suggest that random sequences converge to approximate equilibrium on random networks. As we shall show, there exists a sequence that converges to additive -approximate equilibrium in 1 number of steps that is polynomial in − and the number of edges, in bipartite graphs. The property of graphs that is critically used is that it has no odd cycles, which is an equivalent characterization of bipartite graphs.

Input: Edge e

Modify current state s to a new state s such that xs (u, e)=ys(u, e) and

xs (v, e)=ys(v, e), and s matches s on all edges except e; Function Oracle Update(e)

55 Theorem 4. If the bargaining network and the solution concept satisﬁes the bargain monotonicity condition, then from any start state s, there is a polynomial-time computable se- 2 quence of at most O(m cmax/) edge updates that converge to an additive -approximate bargaining equilibrium, where m is the number of edges. Thus we have an FPTAS if the Update oracle runs in polynomial time.

To get an FPTAS for computing a multiplicative approximate equilibrium (which is an algorithm almost identical to that for the additive approximation), we shall need the Update oracle to further satisfy a basic condition, which is essentially that whenever an edge is updated, none of its endpoints get too small a share.

Condition 3. (Polynomially Bounded Updates Condition) There exists a constant r>0 such that if the outside options of the endpoints of an edge e =(u, v) are at most α, then y (u, e) 1 and y (v, e) 1 for all α 0. s ≥ αr s ≥ αr ≥ Theorem 5. If the bargaining network (and the solution concept) satisﬁes the bargain monotonicity condition and the polynomially bounded updates condition, then from any 2 start state s, there is a polynomial-time computable sequence of at most O(m log ncmax/) edge updates that converge to a multiplicative (1 + )-approximate equilibrium, where cmax is the maximum value of any edge in the network, n is the number of vertices and m is the number of edges. Thus we have an FPTAS if the Update oracle runs in polynomial time.

Before describing the algorithm, we shortly dwell on when and how is the response of the Update oracle computable in polynomial time. It is true for all increasing and continuous utility function in the PBS concept, and also true for all increasing concave functions in the NBS setting. This is because by an application of Lemma 3.2.1 and Lemma 3.2.2 respectively, the problem reduces to being given two functions of the same variable, one increasing and the other decreasing, and being asked to ﬁnd a value of the variable where the two functions are equal. This can be done with exponential accuracy in polynomial time using a binary search process. It is easy to absorb this exponentially small error in the update oracle into the approximation factor of the equilibrium, so we will neglect it. We now proceed to describe our algorithm.

56 3.6.2 The Algorithm

Let s be the current bargaining state, describing division of wealth on each edge. Algorithm 2 describes our algorithm to compute an -approximate equilibrium. For either additive or multiplicative approximation, note that the corresponding deﬁnition for approximate stability of an edge should be used in the algorithm to decide if an edge is -stable.

Input: Bargaining network G and an oracle access to the Update function Output: An -approximate equilibrium Initialize to an arbitrary state s; Color all edges WHITE; while there exists a WHITE edge e do Color e BLACK;

while there exists a BLACK edge e that is not -stable in the current state do Update (e); Output current state; Algorithm 2: An FPTAS for computing an -approximate equilibrium

It is fairly easy to see that when Algorithm 2 terminates, the final state of the bargaining network is an -approximate equilibrium. The outer while loop implies that algorithm can terminate only when all edges are colored black. Moreover, the inner while loop can terminate only when all black edges become -stable. Thus in the last repetition of the outer loop, the last white edge gets colored black, and then the inner while loop ensures that the algorithm terminates only when all the black edges (which is the entire network now) are -stable, and thus the current state is an -equilibrium. So we only need to argue the termination and running time of the algorithm. Since the inner loop of the algorithm terminates only when all the black edges are - stable, so at the beginning of the next inner loop, only the new black edge e may not be -stable. If it is not, the loop terminates without a single call to the oracle. However, if e is unstable, then it is the first edge to be relaxed in this execution of the loop, and the influence of this update now travels along the black edges. Let e =(u, v), and suppose updating e caused u to receive more wealth from the deal on e, than it was receiving just before the update. In such a case, we say that the update favors u. Noting that G is a bipartite graph, we label every vertex in the same partition

57 as u as +, and the vertices in the other partition as . This labeling is only for the sake − of analysis, and may be diﬀerent for distinct executions of the inner loop. Note that every edge is between a vertex labeled + and a vertex labeled . Lemma 3.6.1 is crucial, and − Theorem 4 follows from it. The proof of 3.6.1 depends on Condition 2 being satisﬁed.

Lemma 3.6.1. If Bargaining Monotonicity Condition holds, then during the execution of the inner loop, whenever an edge e =(u,v) is updated, such that u is labeled +,theupdate favors u.

Proof. We shall prove this statement by induction. By definition, the statement holds for the first step, that of updating e. For the inductive step, suppose that the statement has been true for the first i 1 update steps, i 2. Consider the ith update step, where the − ≥ edge updated is e =(u,v). Since the beginning of the loop, whenever an edge incident on u has been updated, by the induction hypothesis, u was favored as it is labeled +. Note that since e is a black edge, it was -stable at the beginning of this loop. So, since the last time that e was -stable, all the updates have only increased the outside option of u with respect to e. By a similar argument, the outside option has gone down or stayed the same for v. Thus Condition 2 implies that u is favored in this update step, and the inductive proof is complete.

Proof of Theorem 4. If we only update edges that are not -stable in the additive sense, then since every update increases the wealth of the favored vertex by at least , and since the wealth from an edge e to any of its endpoint cannot exceed cmax, the edge shall not be updated more than cmax/ times in one iteration of the inner loop. Finally, in each repetition of the inner while loop, at most m distinct edges are updated, and the loop itself is repeated m times. This completes the proof of Theorem 4.

Proof of Theorem 5. Now, suppose we only update edges that are not -stable in the multiplicative sense. Assume that the Polynomially Bounded Updates Condition is true. Then, whenever an edge has been updated at least once before, both of its endpoints receive a share whose inverse is polynomially bounded in ncmax, which is an upper bound on all outside options. Either of these shares can at most go up to cmax within a single iteration of the outer loop, and on every update it goes up (or down) by a factor of at least (1 + )

58 (or (1 )) so the edge cannot be updated more than O( 1 log(nc )) times. Again, in − − max each repetition of the inner while loop, at most m distinct edges are updated, and the loop itself is repeated m times. This completes the proof of Theorem 5.

3.7 Inequality of Sharing in PBS vs NBS equilibria

In this section, we shall study the effect of network topology on PBS and NBS equilibria, and compare the two. Our result shows that we should expect more skewed deals in a PBS equilibrium compared to an NBS equilibrium. In the rest of this section, we shall assume here that all vertices have the same utility function (x), and that the deal on every edge has unit profit, so that the network topology U is solely responsible for any skewness in the distribution of profits in an equilibrium. We also assume some natural properties of the utility function that are satisfied by all functions of the form xp.

3.7.1 Inequality in NBS equilibrium

The theorem below implies that if two adjacent vertices have high (but possibly highly unequal) degrees, then the deal between them is shared almost equally in any NBS equilibrium. We shall later show that this is not true for PBS equilibria.

Theorem 6. Let (x) be the utility function of every vertex, and let all edges have unit U (x) (0) weight. Let (x) be increasing, twice diﬀerentiable and concave. Also, suppose U −(xU) < U U Kx x [0, 1] for some constant K,and (x) (x) (x) for some decreasing function ∀ ∈ |U | ≤ U (x). Let s be any NBS equilibrium in this network. Let e =(u, v) be an edge such that u and v have degree more than (K +1)d+1 for some positive integer d. Then, x (u, e) 1 < (d). | s − 2 | Note that the assumptions on the utility function guarantee the existence of NBS equilibrium. Also note that the function (x)=xp for some 0

59 Lemma 3.7.1. Let (x) be increasing, twice diﬀerentiable and concave. Also, suppose U (x) (0) U −(xU)

Claim 2. If s1 and s2 are any two states such that αs1 (u, e) > αs2 (u, e), then qs1 (x) < q (x) x [0, 1]. s2 ∀ ∈

Proof. To complete the proof, we need to prove our claim. For this, we view qs(x) as a function z(α) of αs(u, e), keeping x constant, where α = αs(u, e), and observe that this (α+x) (α) function is decreasing. That is, z(α)= U (α+−xU) ,wherex is constant. Note that U (α + x) (α) x (α + x). Diﬀerentiating U − U U U − U U z(α)withrespecttoα and using the above inequalities, we get that the numerator of the derivative z (α)is (α + x)( (α + x) (α)) ( (α + x) (α)) (α + x) < 0, while U U − U − U − U U the denominator is positive. So z(α) < 0, which implies our claim.

(x) (0) Thus, qs(x) is greatest when αs(u, e) = 0, in which case qs(x)= U −(xU) 1 x, s s − or x>1/(K + 1). This completes our proof of Lemma 3.7.1.

Lemma 3.7.2. Let (x) be increasing, twice diﬀerentiable and concave. Let s be an NBS U equilibrium, e =(u, v) be any edge, and > 0. Also, let (α (u, e)) (α (u, e)) and |U s | ≤ U s (α (v, e)) (α (v, e)). Then, if u gets x on this agreement at equilibrium (and v |U s | ≤ U s gets 1 x), then x 1 < . − | − 2 |

Proof. Let α = αs(u, e) and β = αs(v, e). According to NBS, x is the solution maximizing ( (α + x) (α))( (β +1 x) (β)). Diﬀerentiating the quantity with respect to x and U − U U − − U equating to zero, we get the following condition: (α + x)( (β +1 x) (β)) (β + U U − − U − U 1 x)( (α + x) (α)) = 0. − U − U

60 We are seeking a solution to this equation. The left-hand-side of the above equation is a function g(x) of the form h (x) h (x) that is decreasing, continuous, positive at x =0 1 − 2 and negative at x = 1. Thus, the equation g(x) = 0 has a solution in x (0, 1). ∈ Note that (β +1 x) (β)liesbetween (β)(1 x) and (β + 1)(1 x), while U − − U U − U − (α + x) (α)liesbetween (α)x and (α + 1)x. Also, (β +1 x) > (β + 1) > U − U U U U − U (β)+ (β), and (α + x) > (α + 1) > (α)+ (α), for all x (0, 1). U U U U U U ∈ Now, for contradiction, suppose x 1 = 1 (1 2). Then, ≤ 2 − 2 −

h (x) > (α + 1) (β + 1)(1 x) > ( (α)+ (α))( (β)+ (β))(1 x) 1 U U − U U U U − (α) (β) = (α) (β)(1 + U )(1 + U )(1 x) U U (α) (β) − U U 1 1 (α) (β)(1 )(1 )( )(1 + 2) > (α) (β)(1 2) ≥ U U − − 2 2U U −

(α) (β)x>h (x) ≥ U U 2 Thus, g(x) > 0 when x 1 . Similarly, g(x) < 0 when x 1 + . ≤ 2 − ≥ 2 Proof of Theorem 6. There are (K +1)d edges incident on each vertex u and v excluding 1 (u, v), so Lemma 3.7.1 implies that at an NBS equilibrium, αs(u, e) > K+1 (K + 1)d = d and α (v, e) > 1 (K + 1)d = d.Since (x) (x) (x) and (x) is decreasing, we put s K+1 |U | ≤ U = (d) < min (α (u, e)), (α (v, e)) in Lemma 3.7.2 to obtain our result. { s s }

3.7.2 Inequality in PBS equilibrium

In contrast, consider two star networks (where leaves are attached to a single central node), one with d1 leaves and the other with d2 leaves, and consider the composite network G formed by adding an edge between the centers of the two stars. Let d1 >> d2 >d. Let x1/2 be the utility function of all players. Then, Theorem 6 implies that the edge between the two centers is shared such that each party gets at least 1 1 . In contrast, it can be 2 − 2d checked that in a PBS equilibrium, the wealth of the two centers are close to d1 and d2 respectively, and the deal between them is shared at a ratio close to √d1 to √d2 for large values of d.Sothediﬀerence in the share of this edge between PBS and NBS equilibrium is stark when, say, d1 = 10000 and d2 = 100 – if the players are sharing a dollar, then the

61 skew is less than a cent for NBS equilibrium, while the center of the star with 100 leaves receives less than 10 cents on the deal in a PBS equilibrium.

3.8 Simulation Studies

While a number of works have strived to quantify relationships between network structure and various game-theoretic equilibria [68, 63, 37, 17], precise characterizations are rare. We turn to an alternative approach, which is that of empirically investigating the structure- equilibrium relationship in networks randomly generated from well-studied stochastic formation models such as preferential attachment (PA), that gives scale-free networks, and Erdos-Renyi random graphs. We are particularly interested in studying how a player’s position in the network inﬂuences its bargaining power, so we assign identical utility functions to all players. For all our simulations, unless otherwise mentioned, we use the utility function (x)=√x. U The summary of our ﬁndings is as follows:

Random best response dynamics on a network always appears to converge to the • same PBS or NBS equilibrium. This provides evidence that these networks have unique equilibrium.

In each random network family there is very high positive correlation between degree • and wealth, for either equilibrium concepts. However, degree alone is not a good predictor of wealth.

Bargaining power of a vertex, measured as the average share received by the vertex • on all its edges, increases with degree.

Outcomes given by the two equilibrium concepts diﬀer strongly. Eﬀects of network • structure is more pronounced in PBS equilibrium. The bargaining powers of vertices have a greater variance in PBS equilibrium. Also, the divisions on edges are generally more skewed in PBS equilibrium and more balanced in NBS equilibrium.

62 3.8.1 Methodology

The broad methodology we followed was to (a) generate many random networks from speciﬁc stochastic formation models (namely, preferential attachment [9] and Erdos-Renyi [48]), (b) compute bargaining equilibria on these networks by running best-response dynamics until convergence, and (c) perform statistical analyses relating structural properties of the network to equilibrium properties. For the best response dynamics, we start from a random state, and then repeatedly pick any edge that is not -stable and update it, until all edges are -stable, for =0.001. In all our simulations, all edges in the network have unit wealth. Also, in all our simulations, we imposed the same utility function on all vertices, so that the sole diﬀerence between U the players is their positions in the network. Unless mentioned otherwise, = √x in our U simulations; on some simulations we chose = xp for various values for 0

63 3.8.2 Correlation Between Degree and Wealth

Echoing earlier results found in a rather different (non-bargaining) model [65], we found that in both formation models there is a very high correlation between vertex degree and wealth at equilibrium — on average (over networks), correlations in excess of 0.95. Given such high correlations, it is natural to attempt to model the wealth of each vertex in a given network by a linear function of its degree — that is, in a given network we approximate the equilibrium wealth w of a vertex v of degree d by αd β, and minimize the mean v v v − squared error (MSE) 1 (w (αd β))2 n v − v − v V ∈ where n = V . We find that such fits are indeed quite accurate (low MSE). We do note | | that the correlation is generally higher, and the MSE generally lower, in NBS equilibrium compared to PBS equilibrium, so linear functions of degree are better models of wealth in NBS than PBS. Note that since the sum of the wealth of all vertices is equal to the number of edges m, we have m αd β m α(2m) βn m β (2α 1) v − ≈ ⇒ − ≈ ⇒ ≈ n − v V ∈ Thus, β˜ = m (2α 1) is an estimate of β that is almost accurate when the mean squared n − error is as small as we have found. β˜ is positive if α > 0.5, and negative if α < 0.5. In essence, therefore, the wealth distribution on the vertices of a specific network is succinctly really expressed by just a single real value – the degree coefficient α. Note that α itself is a function of the network, and as we shall see, is also dependent on the equilibrium concept.

3.8.3 Regression Coeﬃcients

For each graph, we have two coefficient values α, one for the PBS equilibrium and the other for the NBS equilibrium. Thus each distribution of graphs gives two distributions of α, one for PBS equilibrium and the other for NBS equilibrium. A standard t-test reveals that for both network formation models, these distributions of coefficients have rather means at a high level of statistical significance. On all networks, the coefficient in the PBS equilibrium was higher than that in the NBS equilibrium, and both numbers were greater than 0.5.

64 1

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0.9 cient PBS cient

ﬃ 0.75 0.85 ﬃ

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0.65 0.6 Average regression coe Average Regression coe 0.6 NBS NBS 0.55 0.55

0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 BA graphs(50 nodes): New links per vertex ER graphs (50 nodes):Probabilityofanedge

Figure 3.1: PBS and NBS regression coeﬃcients versus edge density in (a) PA and (b) ER networks

Figure 3.1(a) shows the values of α for PBS and NBS equilibria in the 5 different distributions, which vary in their edge density, from the preferential attachment model. The horizontal axis shows the number of new links added per vertex in the random generation, while the vertical axis shows the regression coefficients from 100 trials. Figure 3.1(b) shows the analogous plot for distributions from the Erdos-Renyi model, with the horizontal axis representing the edge probability. The plots clearly demonstrate that the dependence of wealth on degree is stronger in PBS than in NBS. In preferential attachment networks, both equilibrium concepts seem to have diminished coefficients with increased edge density, but this effect is absent or muted in Erdos-Renyi.

3.8.4 Division of Wealth on Edges

So far we have examined the total wealth of players in equilibrium states; we can also examine how the wealth on individual edges is divided at equilibrium. Figure 3.2 shows histograms of the division of wealth on edges in the distribution PA(50, 2) for PBS and NBS equilibria. The horizontal axis shows the amount of the smaller share of an edge, and the vertical axis shows the number of edges whose smaller share is within the given range. The number of edges is summed over 100 graphs from each distribution. In NBS, the wealth on most edges are split quite evenly, while in PBS, the split is heavily skewed, yet another indication that network structure plays a greater role in PBS equilibrium than

65 1400 5000

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0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

Figure 3.2: A histogram of PBS and NBS equilibria, respectively, for inequality of each edge, for PA(50, 2) in NBS equilibrium. A more reﬁned view of this phenomenon is provided in Figure 3.3. Here we show the average edge divisions as a function of the degrees of the two endpoints d1 and d2 for the two equilibrium notions in the distribution PA(50, 3). We note that while both surfaces are smooth and have similar trends, the slope of the surface is much gentler in NBS equilibrium than in PBS equilibrium, demonstrating that even neighbors with rather diﬀerent degrees tend to split deals approximately evenly at NBS equilibrium.

3.8.5 Other Utility Functions

All experiments described so far examined the utility function √x. We also performed experiments examining equilibria varied with a change of utility functions. We examined = xp where p = i/10, 1 i 10, on each graph distribution, and found that for U ∀ ≤ ≤ each of them, the correlation of wealth and degree was still very high and linear ﬁts still provide excellent approximations. We know theoretically that for i = 10, that is, (x)=x, U we have α =0.5. Figure 3.4(a) illustrates how the degree coeﬃcient for PBS equilibrium decreases smoothly in PA(50, 3), from an average value of almost 1 to 0.5, as p goes from 0.1to1.0, while that for NBS equilibrium starts barely above 0.5 and also goes down to 0.5, albeit with a far gentler slope. Figure 3.4(b) shows the same plot for the distribution ER(50, 0.12), which, in expectation, has approximately the same number of edges as in

66 1

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0.4 0.5 0.3 0.45 0.2 0.4 0.1 0.35 0 30 Share received by the vertex on edge with neighbor 30 25 3025 20 25 30 20 25

15 20 Share received by the vertex on edge with neighbor 15 20 15 10 10 15 Degree of the vertex 10 10 5 Degree of neighbor Degree of vertex 5 Degree of the neighbor 5 5 0 0 0 0

Figure 3.3: Edge division versus edge endpoint degrees in PBS and NBS, resp., for PA(50, 3)

PA(50, 3). Thus, again viewing a higher value of α as a higher variance in bargaining power and thus greater effects of network structure, with α =0.5 implying the absence of network effect, we conclude from the figures that network effects gradually diminish when the utility function approaches linearity.

1.1 1

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0.5 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Exponent of utility function Exponent of utility function

Figure 3.4: Regression coeﬃcient against exponent of utility function, in (a)PA(50, 3) and (b)ER(50, 0.12) resp.

67 3.9 Uniqueness of Equilibrium

The simulations suggested that equilibrium may be unique in the networks we ran our simulations on, or at least that the random best response dynamics converges to a unique one. In this section, we address the question of whether equilibrium is unique. Unfortunately, the answer to this question is no. We focus on regular graphs with unit wealth on all edges, and the same utility function , which is increasing, continuous and concave, for all vertices. This class has one state U that is both PBS and NBS equilibrium: the state where the value on every edge is divided into two equal parts. In fact, one may consider this state to be the natural solution on regular networks. We investigate if there is any other equilibrium. We show that one can choose such that there are multiple equilibria, both for PBS and for NBS. U However, we give a simple condition of the update process that will ensure uniqueness in this class of networks. We show that many natural concave utility functions such as xp, 0

3.9.1 PBS Equilibrium is Not Unique

Consider any d-regular bipartite graph (d 2) with edges of unit wealth, and every player ≥ with the same utility function , which is defined as (x) = 100x for 0 x 0.01, and U U ≤ ≤ (x) = log x log 0.01 + 1 for x>0.01. Then, (0) = 0, is differentiable, and is a U − U U U decreasing positive function, so is concave and increasing. One may view as a shift of U U the logarithmic function designed to ensure that concavity and the condition (0) = 0 are U satisfied. For this utility function and any regular network, there are uncountably many PBS equilibria. Let the vertices in the bipartite graph be X Y ,whereX and Y are independent sets. ∪ Consider a state s where on every edge, the endpoint in X receives 1/2 , while the endpoint − in Y receives 1/2+, for any 0 < < 0.49. Then, for any edge, the outside options of its two endpoints are (d 1)(1/2 ) and (d 1)(1/2+) respectively, and s is a PBS equilibrium − − − if and only if (d(1/2 )) ((d 1)(1/2 )) = (d(1/2+)) ((d 1)(1/2+)). U − − U − − U − U − It is easy to check that both sides evaluate to log d/(d 1). Since the result holds for any −

68 0 < < 0.49, we have a continuum of PBS equilibria.

3.9.2 NBS Equilibrium is Not Unique

We again consider d-regular bipartite graph (d 2) with edges of unit wealth, and every ≥ player with the same utility function . However, we need to choose more carefully, and U U we shall only get multiple NBS equilibrium, instead of uncountably many. Let the vertices in the bipartite graph be X Y ,whereX and Y are independent sets. ∪ Consider a state s where on every edge, the endpoint in X receives 1/4, while the endpoint in Y receives 3/4. Then, for any edge, the outside options of its two endpoints are (d 1)/4 − and 3(d 1)/4 respectively, and s is an NBS equilibrium if and only if − (d/4) ((d 1)/4) (3d/4) (3(d 1)/4) U − U − = U − U − (d/4) (3d/4) U U Note that 3(d 1)/4 >d/4 when d 2, so the above equation is easy to satisfy. We can − ≥ deﬁne an increasing, continuous and concave function such that (d/4) = 1, (3d/4) = U U U 1/2 and ( (d/4) ((d 1)/4)) = 2( (3d/4) (3(d 1)/4)). In particular, we can choose U −U − U −U − such that =8x x (d 1)/4, so that ((d 1)/4) = 2(d 1), and then decrease the U U ∀ ≤ − U − − slope gradually such that (d/4) = ((d 1)/4) + 1 = 2d 1 and (d/4) = 1, and then U U − − U (3(d 1)/4) = 1, and ﬁnally (3d/4) = (3(d 1)/4) + 1/2 and (3d/4) = 1/2. U − U U − U

3.9.3 Uniqueness in Regular Graphs

The following condition, which is dependent on the utility function of the players as well U as the concept we are considering, PBS or NBS, is suﬃcient to ensure uniqueness on regular graphs with unit wealth on edges and identical utility functions for the players. We call it a rallying condition, because it allows the player with less outside options to bargain a greater share than what the disbalance in the outside options suggest, even though it gets the smaller portion.

Condition 4 (Bargain Rallying Condition). Let s be any state of a bargaining network and e =(u, v) be any edge of wealth c. Without loss of generality, suppose α (u, e) α (v, e). s ≤ s cαs(u,e) Then, our condition is that c/2 ys(u, e) > . ≥ αs(u,e)+αs(v,e)

69 This condition is satisﬁed by common concave utility functions such as (x)=xp where U 0 0,b>0. A detailed discussion is U deferred to Section 3.9.4. We shall now present and prove our main uniqueness result.

Theorem 7. If Condition 4 is satisﬁed in the PBS (or NBS) concept on a regular graph with unit wealth on every edge and identical utility function for all players, which is increasing, continuous and concave, then there is a unique PBS (or NBS, respectively) equilibrium, where the wealth on every edge is shared equally between its endpoints.

Proof. By contradiction. Clearly, the state, where wealth on every edge is equally divided, is an equilibrium. Suppose there exists another equilibrium s. There exists an edge where xs(u, e) < 1/2 or xs(v, e) < 1/2. Consider the edge e =(u, v) which has the most lopsided division, that is, ﬁnd e such that min (xs(u, e),xs(v, e)) is minimized. Without loss of generality, suppose that u gets the smaller share 1/2 on e, for some > 0. Since − Condition 4 is satisﬁed, so α (u, e) α (v, e). Note that u gets at least 1/2 from each s ≤ s − edge incident on it, and so α (u, e) (d 1)(1/2 ). Similarly, v gets at most 1/2+ s ≥ − − 1/2+ from each incident edge, so αs(v, e) (d 1)(1/2+). Thus αs(v, e)/αs(u, e) 1/2 . Now ≤ − ≤ − 1 using Condition 4, we get that ys(u, e) > 1/2+ =1/2 which is a contradiction to our 1+ 1/2 − assumption that s is an equilibrium. −

3.9.4 Characterization of Bargain Rallying Condition

We try to recognize simpler easy-to-check conditions on that is suﬃcient for the bargain U rallying condition to hold in the PBS or NBS concept on networks with utility function U for all players. We always assume that is increasing, continuous and concave. First, we U consider the PBS concept.

Condition 5. Let be a utility function. Our condition is that (Kx) (x) is an U U − U increasing function of x, x>0, K>1. ∀ ∀ Lemma 3.9.1. If all players have the same utility function that satisﬁes Condition 5, U then Condition 4 holds in the PBS concept.

70 Proof. Let a = αs(u, e) and let b = αs(v, e), and let a0,b>0, satisfy Condition 5. To U illustrate, we show the computation for (x)=xp: (Kx) (x)=Kpxp xp = xp(Kp 1). U U −U − − Note that (Kp 1) is a positive constant while xp is an increasing function of x. − Following the same thread of reasoning, we can ﬁnd a similar condition for NBS.

(Kx) (x) Condition 6. Let be a utility function. Our condition is that U (Kx−U) is an increasing U U function of x, x>0, K>1. ∀ ∀ Lemma 3.9.2. If all players have the same utility function that satisﬁes Condition 6, U then Condition 4 holds in the NBS concept.

Proof. The proof is almost identical to that of Lemma 3.9.1, except that we apply Lemma 3.2.2 instead of Lemma 3.2.1. We deﬁne ((1 + c )x) (x) g(x)=U a+b − U ((1 + c )x) U a+b and by a similar argument, Condition 4 holds if g(x) is increasing, which is satisﬁed if (Kx) (x) U (Kx−U) is increasing in x when K>1. U It is easy to verify that common concave utility functions such as (x)=xp where U 0 0,b>0, satisfy Condition 6. To U illustrate, we show the computation for (x)=xp. U

p p (Kx) (x) (K 1)x 1 1 p 1 p p p 1 1 p p U − U = − = p− K − x − (K 1)x = p− K − (K 1)x (Kx) pKp 1xp 1 − − U − − p 1K1 p(Kp 1) is a positive constant while x, of course, is an increasing function of x. − − − (Kx) (x) So U (Kx−U) is an increasing function of x. U 71 3.10 Generalized Utility Functions

In all previous results in this chapter, as well as the definitions in the Preliminary section, utility function was assumed to be a function of the sum of the wealth received by the player on all edges incident upon it, that is the function was of the form (x + x ...+ x ), U 1 2 d where x1,x2 ...xd are the different shares it receives from its d edges. We can, instead, define a more general concept of a utility function, one that is multi-dimensional, and is an arbitrary function of the values of the wealth it receives from its edges, that is, of the form (x ,x ...x ). The concepts of PBS and NBS equilibrium can be easily extended U 1 2 d to this setting, by simply redefining difference utility, which is still the additional utility a player receives from the deal, given what it is getting from its other deals. In particular, its difference utility from the first incident edge is (x ,x ,x ...x ) (0,x ,x ...x ). U 1 2 3 d − U 2 3 d Note that the outside option for this first edge should also be redefined as a sequence of d values (0,x2,x3 ...xd) where the value corresponding to the first edge is zero, while the rest reflects the wealth the player receives from its other (d 1) deals. − We think it is worth noting that even under this generalized model, the existence results of [24] hold, that is,

PBS equilibrium exists on all networks if the utility functions are increasing and • continuous.

NBS equilibrium exists on all networks if the utility functions are increasing, contin- • uous and concave.

These results follow from the fact that analogous versions of Lemmas 3.2.1 and 3.2.2 hold in this model too, using the fact that an increasing, continuous and/or concave multi- dimensional function has the same property along each dimension. Further, it is easy to verify that the bargaining monotonicity condition is satisfied by concave utility functions in the PBS concept. Again, the condition needs to be modified to take care of the fact that outside options are now a sequence: we say that α (u, e) α (u, e) s ≥ s if the former dominates the latter in every dimension. This condition is sufficient to show that the same FPTAS algorithm works even in this generalized model.

72 3.11 Conclusion

Our model is an addition to the extensive literature on network exchange theory, and it differs from previous models in that network effects are caused by the non-linearity of utility functions of the players. Even when players can enter into as many deals as their degree, our model still suggests a network effect on outcomes and unequal sharing of deals. Further, the effect of network structural properties such as degree on the outcome propagates beyond the immediate neighborhood. The most prominent theoretical question left unanswered in our model is that of computing approximate or exact equilibria in general graphs, and to find precise conditions on utility functions that guarantee uniqueness of equilibrium.

73 Chapter 4

Item Pricing in Combinatorial Auctions

In this chapter, we begin our study of revenue maximizing strategies for a single seller in the presence of many buyers. Results in this chapter were published in collaboration with Zhiyi Huang and Sanjeev Khanna [22]. We consider the following Item Pricing problem. There are n items owned by a single seller, and m prospective buyers who value these items. We assume a limited supply setting where each item can be sold to at most one buyer.The seller can price each item individually, and the price of a set of items is simply the sum of the prices of the individual items in the set. The buyers arrive in a sequence, and each buyer has her own valuation function v(S), deﬁned on every subset S of items. Each buyer chooses a set of items to buy (depending on the prices), and these items become unavailable to future buyers. We assume the valuation functions to be subadditive, which means that v(S)+v(T ) v(S T ) for any pair of subsets ≥ ∪ S, T of items. For some results, we shall assume the valuations to be XOS, that is, they can be expressed as the maximum of several additive functions. XOS functions are subadditive, and include all submodular functions. The strategy used by the seller in choosing the prices of the items is allowed to be randomized, and is referred to as a pricing strategy. If a buyer buys a subset of items S,her utility is deﬁned as her valuation v(S) of the set minus the price of the set S (sum of prices of individual items in S). We assume that every buyer is rational, and so buys a subset

74 of available items that maximizes her utility. We assume that a buyer cannot strategize over her position in the arrival sequence of buyers – she comes to the seller immediately when she has a demand for the items, and cannot advance or postpone her arrival time deliberately, neither can she come to the seller more than once (possibly as a reaction to the seller’s pricing strategy). The revenue obtained by the seller is the sum of the amounts paid by each buyer, and our goal is to design pricing strategies that maximize the expected revenue of the seller. In this chapter, we design strategies in a setting where the seller has no knowledge of the valuation functions of the buyers, and look for strategies that provide worst-case revenue guarantee on any instance, the only assumption being that the buyers’ valuation functions are subadditive. In other words, we design pricing strategies in a prior-free world.

Pricing Strategies A uniform pricing strategy is one where at any point of time, all unsold items are assigned the same price. The seller may set prices on the items initially and never change them, so that cost of an (unsold) item is the same for every buyer. We call such a strategy to be a static pricing strategy. Static pricing is the most widely applied pricing scheme till date. Alternatively, a seller may show different prices to different buyers (without knowing the buyer’s valuation function) – we shall call this a dynamic pricing strategy. Dynamic strategies are applicable in places where the seller may frequently adjust her prices over time, and are commonly observed while buying airline tickets or booking hotel rooms. However, a dynamic strategy in which the price of an item fluctuates a lot may not be desirable in some applications. However, a dynamic strategy in which the price of an item fluctuates a lot may not be desirable in some applications. So we introduce an interesting subclass of dynamic strategies, called dynamic monotone pricing strategies, where the price of an item can only decrease with time.

Social Welfare An allocation of items involves distributing the items among the buyers, and the social welfare of an allocation is deﬁned as the sum of the buyers’ valuations for the items received by each of them. We denote the maximum social welfare, achieved by any allocation, by OPT. We measure the performance of a pricing strategy as the ratio of the maximum social welfare against the smallest expected revenue of the strategy, for any

75 adversarially chosen ordering of the buyers. (Some of our results, where it will be explicitly stated, shall consider expected revenue under the assumption that the order in which buyers arrive is uniformly random.) If this ratio is at most α on any instance (where α can depend on the size of the instance), we say that the strategy achieves an α-approximation. Note that the maximum social welfare is an upper bound on the revenue the seller can obtain under any circumstance. In fact, there exists simple instances with n items and a single buyer where the maximum social welfare is log n, but the revenue can never exceed 1 for any pricing function [8]. Thus we are comparing the performance of our strategies against a bar that may be signiﬁcantly higher than the revenue of an optimal truthful auction, and the above example shows that we can never hope to achieve anything better than a logarithmic approximation. Our general goal is to design pricing strategies that achieve polylogarithmic approximation.

Related Work A relevant body of literature is the design of truthful or incentive compatible mechanisms for combinatorial auctions [36], where all buyers report their valuations before the seller decides on allocation of the items and prices charged to the buyers. While there exists a truthful mechanism which maximizes social welfare among buyers in all instances, namely VCG auction, there is no such analogue for revenue maximization, unless one makes Bayesian assumptions about buyers’ valuations. Even for welfare maximization, VCG auction may not be efficiently implementable, and this has inspired a large body of work in algorithmic mechanism design that design approximately welfare maximizing truthful mechanisms that are also computationally efficient (see [16] for a survey, as well as [71, 69, 46, 43]). Feige [49] gave a 2-approximation algorithm for computing a social welfare maximizing allocation, but it is not known how to construct a truthful mechanism with such a performance guarantee. Dobzinski et al. [46, 43] gave logarithmic approximation via truthful mechanisms when buyers have XOS valuations and subadditive valuations respectively. The mechanism achieving this approximation is in fact a static uniform pricing strategy. Relatively less work exists on revenue optimization via truthful mechanisms, especially in prior-free settings. Most results assume unlimited supply of copies of each item [59], so that one buyer taking an item affect its availability to future buyers. Pricing strategies

76 form a natural class of truthful mechanisms, and thus the design of such strategies can be viewed in the context of designing truthful mechanisms. Moreover, they are simpler from an organizational point of view, compared to running an auction such as VCG – buyers need not be present in the market at the same time, instead they can come at any time, interact with the seller and leave with a subset of items mutually agreed upon. This is a crucially needed property for many consumer markets, since we do not expect to participate in an auction every time we go to buy milk from a grocery store, or when visiting a garage sale. Setting prices on subsets of items is a mechanism of intermediate complexity, which has many of the advantages of pricing over auctions, but may still be too complex to use, especially if the bundles are priced unnaturally. Item pricing can be viewed as a simple mechanism in this class where the bundles of items are priced additively, and is most natural from a buyers’ perspective. Some research has been directed towards developing truthful mechanisms that approximately maximize revenue [7, 50, 57], while others have focused on designing item pricing strategies that maximizes revenue (eg. [58, 6, 3, 19, 40]). All the pricing strategies analyzed in these papers are static, and often uniform. Most of these results are also in a prior-free model. Under Bayesian assumptions (i.e. buyer valuations are drawn from distributions that are known to the seller), optimal revenue maximizing mechanisms, such as that given by Myerson [76], have been characterized for single parameter settings (such as the case where the seller has only one item to sell), and such characterizations do not apply to multi- parameter settings such as ours. Pricing strategies for revenue maximization is also a commonly studied problem in revenue management (see [85] for a comprehensive overview), with varying degrees of structural assumptions on seller’s inventory and buyers’ valuations and strategic behavior. Most of the work in the revenue management literature makes Bayesian assumptions, and studies complex (i.e. non-uniform, dynamic) pricing strategies that are natural for many applications. Balcan, Blum and Mansour [8] considered static pricing strategies with the objective of revenue maximization, with limited supply of inventory and subadditive buyer valuations, in a prior-free model. In the unlimited supply setting, they designed a pricing strategy that achieve revenue which is logarithmic approximation to the maximum social welfare even

77 for general valuations. The strategy, again, was a uniform strategy. This result was also proved independently in [18]. However, in the limited supply setting, they could only get a 2O(√log n log log n) factor approximation using a static uniform strategy. Crucially, they ruled

1/4 out the existence of static uniform strategies that achieve anything better than a 2Ω(log n) approximation, even if the buyer valuations are XOS, and the ordering of buyers is assumed to be chosen uniformly at random. Thus their result distinguished the limited and unlimited supply settings. This impossibility of getting a good (polylogarithmic) approximation is a consequence of being restricted to static uniform strategies, and it remains impossible even if the seller knew the buyer valuations, and had unlimited computational power. Further, almost all efficient mechanisms designed in these related problems have only used a single price for all items. It is, therefore, natural to consider dropping one of these restrictions, namely, look at dynamic uniform strategies and static non-uniform strategies, both of which use multiple prices, and attempt to find better guarantees on the revenue. It is worth noting that there are other revenue maximizing auctions which give polylogarithmic guarantees, Hartline [60] notes that running the VCG auction with a random uniform offset yields O(log n) approximation for arbitrary valuation functions, though it is not computationally efficient. Also, independent from our work, Dobzinski [42] has developed a bundle pricing strategy for combinatorial auctions with subadditive valuations that gives O(log2 n) approximation – the strategy not only prices items, but also restricts buyers from buying a nonempty subset of items smaller than some critical size. Our results below show that pricing bundles of items is not needed to achieve such a guarantee on revenue. Recently, constant factor approximation results in revenue maximization via item pricing have been obtained in Bayesian models, for special cases of subadditive valuations such as budgeted additive [13] and unit-demand [28] valuations.

Our Results and Techniques The table below summarizes our results on the Item Pricing problem in the limited supply setting, along with relevant earlier work. Our contributions are labeled with the relevant theorem numbers. Our ﬁrst contribution is to strengthen the lower bound results for static uniform pricing, by constructing instances with XOS valuations where uniform pricing functions cannot achieve any better than a 2Ω(√log n) approximation, even if the seller knew the buyer val-

78 Pricing Subadditive valuations -XOS valuations

Strategy Algorithm a Lower Bound b Algorithm a Lower Bound b

Dynamic 2 2 2 log n 2 log n O(log n) Ω log log n O(log n) Ω log log n Uniform [Thm. 9] c [Thm. 10] [Thm. 9] c [Thm. 10] Pricing

Dynamic 2 2 2 log n 2 log n Monotone O(log n) Ω log log n O(log n) Ω log log n Uniform [Thm. 11] d [Thm. 10] [Thm. 11] d [Thm. 10]

Pricing

Static 2O(√log n) 2Ω(√log n) 2O(√log n) 2Ω(√log n) Uniform BBM08[8] c [Thm. 8] BBM08[8] c [Thm. 8] Pricing

Static Non- 2O(√log n) Ω (log n) O(m log log3 n) Ω (log n) Uniform BBM08[8] c BBM08[8] [Thm. 13] c [BBM08 [8]] Pricing

aAll algorithms assume that the seller knows OPT up to a constant factor. This assumption can be removed by worsening the approximation ratio by a factor of log OPT(log log OPT)2. bAll lower bounds are in the full information setting, where the seller knows the buyers’ valuations, the number and arrival order of buyers, has unbounded computational power, and can even force the arrival order of buyers. cBuyers arrive in an adversarial order. Thus the algorithm satisﬁes the upper bound for any order of buyers, including the order that minimizes expected revenue. dBuyers arrive in a uniform random order, that is, every permutation of buyers is equally likely. The bound is on the expected revenue under this assumption. This algorithm also assumes that the seller knows the number of buyers m up to a constant factor. The assumption can be removed by making the algorithm randomized, and worsening the approximation ratio by a factor of log m(log log m)2.

Table 4.1: Summary of results on revenue maximizing item pricing in combinatorial auctions

79 uations, and had unlimited computational power. Our basic construction with only two buyers and a chosen ordering between them, is structurally very similar to the one given in [8], except for a better tuning of parameters. However, this basic construction, as well as that in [8], fails to give a lower bound if we switched the arrival order of the two buyers. In other words, the lower bound of the basic construction holds only if the buyer arrival order is adversarial. We build instances derived from the basic construction to demonstrate how truly limiting static uniform pricing is for combinatorial auctions with limited supply. First, we increase the number of buyers and construct valuations such that the same lower bound holds for uniformly random arrival order of buyers. We further extend the construction to show that the lower bound holds even if the seller could force an arrival order among buyers. All these constructions have simple 3-XOS valuations for all buyers, i.e. XOS valuations that can be expressed as the maximum of 3 additive component functions. Finally, we extend the last instance so that all buyers have identical XOS valuations, though not 3-XOS, and the lower bound continues to hold for any ordering of buyers. Theorem 8 summarizes our lower bound results on static uniform pricing. In contrast, we design a simple randomized dynamic uniform pricing strategy such that its expected revenue is O(log2 n) approximation of the optimal social welfare, when the valuation functions are subadditive, provided the seller knows an estimate of OPT. The strategy randomly chooses a reserve price at the beginning, and then in each round, randomly chooses a price above the reserve price (but less than OPT) and puts this price on each unsold item. By using a fresh random price (from a suitable set of prices) in each round, we guarantee, in expectation, to collect a large fraction of the revenue that can be obtained from the current buyer given the remaining items. Intuitively, an appropriate reserve price helps to avoid selling large number of items cheaply to some buyers before the arrival of a buyer who would be willing to pay a high price. As in [8, 46], all random prices are chosen on the logarithmic scale; in particular, the reserve price may be chosen uniformly at random from the set of powers of 2 (or any constant greater than 1) within the range [OPT, OPT/2n]. The dynamic uniform pricing strategy described above achieves a high revenue, but requires random ﬂuctuation in the price of an unsold item. This may not be a desirable

80 property in some applications. We design a dynamic monotone uniform pricing strategy where the price of any unsold items only decreases over time. We show that if the ordering of buyers is assumed to be uniformly random, that is, all permutations of buyers are equally likely, then the expected revenue of this strategy is an O(log2 n) approximation of the optimal social welfare. The strategy is in fact deterministic, provided the seller knows estimates of OPT and m up to a constant factor, and simply decreases the price gradually over the sequence of buyers. Deterministic strategies giving good approximation in such limited information settings are rare. The usual need for randomness in the face of limited information is taken care of by exploiting the randomness of the arrival order. We emphasize here that our lower bound for static uniform pricing holds for any ordering of buyers. We also construct a lower bound instance to show that the performance of our dynamic uniform pricing strategies is almost optimal among all dynamic uniform strategies. We show that no dynamic uniform pricing scheme can guarantee a revenue of ω(OPT(log log n)2/ log2 n) even for XOS (which is a subclass of subadditive) buyer valuations. This bound holds even when the seller knows the buyers’ valuation functions and unbounded computational power, and can even choose the arrival order among the buyers. All our algorithms as well as the algorithms in [8] assume that OPT is known to the seller up to a constant factor. Moreover, our dynamic monotone strategy assumes that the number of buyers m is known to the seller up to a constant factor. Once this information is provided, the strategies are independent of the buyer valuations, and can be easily seen to be truthful. As Balcan et al. [8] pointed out, for any parameter that is assumed to be known up to a constant factor, if the seller instead knows an upper bound of H and a lower bound of L on the optimum, then this assumption can be removed by guessing the parameter to be a power of 2 between H and L, uniformly at random, worsening the approximation ratio by a factor of Θ(log(H/L)). Note that an estimate of up to a factor of poly(n)suﬃces for our results – as such, an estimate on the highest value of a single item to any one buyer would serve as an adequate estimate. It is not hard to see that even for a single item, the approximation guarantee cannot be better than Θ(log(H/L)) – see [5] for such results even in Bayesian models. If the seller instead knows that OPT 1, but knows no upper bound, ≥ then the assumption can be removed by worsening the approximation ratio by a factor of

81 Θ(log x(log log x)2) in the approximation, where x is the said parameter. Finally, we give a static non-uniform strategy that gives an O(m log log3 n)-approximation if the buyers’ valuations are XOS valuations that can be expressed as the maximum of additive components. Note that when the order of buyers is adversarial, the hard instance for static uniform pricing has only two buyers, and their valuation functions are the maximum of o(log n) additive functions components. In particular, our strategy gives polylogarithmic (in n) approximation when the number of buyers are small (polylogarithmic in n), and has XOS valuations which are the maximum of quasi-polynomial (in n) additive components. We note that these guarantees are quite weak and quite easy to achieve if buyer arrival order is assumed to be uniformly random, or even if the seller is allowed to reject certain buyers (in particular, reject all but one buyer chosen at random, and use the result of Bal- can et. al. [8]). Nevertheless, developing a static non-uniform strategy is an ideal goal for a seller, and can be most widely applicable, since it does not discriminate between buyers at all. Moreover, all lower bounds have so far been proved for uniform strategies only, so non-uniform strategies may even yield better approximation results. The techniques used to prove our result may be useful towards ﬁnding a static non-uniform strategy with better revenue guarantee, so we include it in this work.

Organization The rest of the chapter is organized as follows. In Section 4.1, we formally describe the problem, establish our notations and present some preliminary lemmas that will be useful in proving our main results. In Section 4.2, we present our lower bound results for static uniform strategies. In Section 4.3, we present our algorithms as well as lower bound for dynamic uniform strategies. In Section 4.4, we present our results for static uniform strategies. We discuss our conclusions and some open problems in Section 4.5.

4.1 Preliminaries

In the Item Pricing problem, we are given a single seller with a set I of n items that she wishes to sell. There are m buyers, each with their own valuation function deﬁned on all subsets of I. A buyer with valuation function v values a subset of items S I at v(S). ⊆ The buyers arrive in a sequence, and each buyer visits the seller exactly once. The seller

82 is allowed to set a price on each item, and the price of a subset of items is the sum of the prices of items in that subset. For every item sold to the buyers, the seller receives the price of that item. Note that an item can be sold at most once. So a seller can only offer those items to a buyer that has not been sold to any previous buyer. The revenue obtained by the seller is the sum of the prices of all the sold items. Each buyer buys a subset of the items shown to her that maximizes her utility,whichis defined as the value of the subset minus the price of the subset. This is clearly the behavior that is most beneficial to the buyer. The Item Pricing problem is to design (possibly randomized) pricing strategies for the seller that maximizes the expected revenue of the seller. Unless noted otherwise, all our algorithmic results will assume that the seller has no knowledge of the order of arrival of the buyers, total number of buyers, or the valuation functions of buyers. We refer to a setting as the full-information setting if all these parameters are known to the seller.

Valuation Functions Throughout this chapter, we will assume that the buyer valuation function v is subadditive, which means that

v(S)+v(T ) v(S T ) , S I,T I. ≥ ∪ ∀ ⊆ ⊆

Unless explicitly stated otherwise, this will be the only assumption on the buyer valuation functions. For some results, we shall assume the buyer valuations to be more restrictive than subadditive.

Deﬁnition 4.1.1. A subadditive valuation function v is called an XOS valuation if it can be expressed as v(S) = max a (S),a (S),...,a (S) on all subsets of items S I, where { 1 2 } ⊆ a1,a2 ...a are non-negative additive functions. The functions a1,a2,...,a are referred to as the additive valuation components of the XOS valuation v.Wesaythatv is an -XOS function if it can be expressed using at most additive valuation components.

We note that a 1-XOS function is simply an additive function, that all XOS valuations are subadditive, and that not all subadditive valuations can be expressed as XOS valuations [70].

83 Pricing Strategies We will study the power of some natural classes of pricing strategies.

Deﬁnition 4.1.2. A pricing strategy is said to be static if the seller initially sets prices on all items, and never changes the prices in the future. A pricing strategy is said to be dynamic if the seller is allowed to change prices at any point in time. A dynamic pricing strategy is also said to be monotone if the price of every item is non-increasing over time.

Deﬁnition 4.1.3. A pricing strategy is said to be uniform if at all points in time, all unsold items are assigned the same price.

4.1.1 Notation

For a buyer B with a valuation function v,weuseΦ(B,J,p) to denote a set of items that the buyer B may buy when presented with set J of items, each of which are priced at p.Since v(S) p S is the utility if the buyer buys the set S,soΦ(B,J,p) argmaxS Jv(S) p S − | | ∈ ⊆ − | | maximizes the utility. Note that there may be multiple possible sets that maximize the utility. In this chapter, when we make a statement involving Φ(B,J,p), the statement shall hold for any choice of these sets. We shall denote the maximum utility as (B,J,p); note U that in contrast to Φ(B,J,p), the value (B,J,p)isuniquelydefined.Whentheunderlying U buyer B is clear from the context, we shall denote these two values as Φ(J, p) and (J, p) U respectively. Moreover, if the set of available items J is also clear from the context, then we shall denote these two values as Φ(p) and (p) respectively. For any set S and a buyer U with valuation function v,wedefineHv(S) = maxS S v(S) as the maximum utility the ⊆ buyer can get if all items in S are offered to her at zero price.

Deﬁnition 4.1.4. We say that a set of items S is supported at a price p with respect to some buyer B with valuation function v,ifB buys the entire set S when the set S is presented to B at a uniform pricing of p on each item.

The following lemma follows easily from the fact that the valuation functions are subadditive, and appears in earlier works as well (e.g.. [43, 8]). Its proof is included for the sake of completeness.

Lemma 4.1.1. Let S be a set of items that is supported at price p. with respect to a buyer B with subadditive valuation function v. Then v(S ) p S for all S S. ≥ | | ⊆ 84 Proof. Suppose not. Then there exists S S with v(S )

v(S) v(S) p S S p S + p S =(v(S) p S ) v(S)+p S − − | \ | − | | | | − | | − | | >v(S) p S , − | | contradicting the assumption that buyer picks set S at price p.

4.1.2 Optimal Social Welfare and Revenue Approximation

We now deﬁne optimal social welfare, the measure against which we evaluate the performance of our pricing strategies.

Definition 4.1.5. An allocation of a set S of items to buyers B1,B2 ...Bm with valuations v ,v ...v , respectively, is an m-tuple (T ,T ,...,T ) such that T S for 1 i m,and 1 2 m 1 2 m i ⊆ ≤ ≤ T T = for 1 i, j m.Thesocial welfare of an allocation is defined as m v (T ),and i∩ j ∅ ≤ ≤ i=1 i i m an allocation is said to be a social welfare maximizing allocation if it maximizes i=1 vi(Ti). The optimal social welfare OPT is defined as the social welfare of a social welfare maximizing allocation.

Clearly, OPT is an upper bound on the revenue that any pricing strategy can get. Let R be the revenue obtained by the strategy, which is the sum of the amounts paid by all the buyers.

Deﬁnition 4.1.6. A pricing strategy is said to achieve an α-approximation if the expected revenue of the strategy E[R] is at least OPT/α.

Unless stated otherwise, expected revenue is computed assuming an adversarial ordering of the buyers, that is, the ordering that minimizes the expected revenue of the strategy. In other words, we require a strategy to work well irrespective of the order of buyers in which they arrive. Note that OPT is not a tight upper bound on the maximum revenue that can be achieved by any pricing strategy, even with full knowledge of buyer valuations and unbounded computational power. In fact, the following example appears in Goldberg et al. [61] and Balcan

85 S et al. [8]: if there is a single buyer with valuation function v(S)= i|=1| 1/i, then for any pricing of the n items, the revenue is at most 1, while OPT = Θ(logn). This shows that nothing better than a logarithmic approximation can be achieved in the absence of any other assumption on the buyer valuations.

4.1.3 The Single Buyer Setting with Uniform Pricing Strategies

Balcan et al. [8] considered the setting where there is an unlimited supply of each item, so that no buyer is aﬀected by items bought before her arrival. In particular, if there is only a single buyer, then there is no distinction between limited and unlimited supply, as long as the buyer never wants more than one copy of the same item. For the single buyer case, Balcan et al. [8] gave an O(log n) approximation, and in the process proved some lemmas that will be useful for our algorithmic results in the limited supply setting as well. Suppose a set S is being shown to a buyer B with valuation function v. The optimal

social welfare in this single buyer instance is Hv(S). We consider setting a uniform price, that is, the same price on all items. The following lemma (which also appeared in Balcan et. al. [8]) states that the number of items bought monotonically decreases as the price on the items is increased.

Lemma 4.1.2. (Lemma 6 of [8]) Suppose a buyer B is oﬀered a set S of items using a uniform pricing. Then for any p>p 0,ifB buys Φ(p) if all items are priced at ≥ p,andΦ(p ) if all items are priced at p , then Φ(p) Φ(p ) . Thus there exist prices | | ≤ | | = q >q >...>q >q =0and integers 0=n

(p)= (q )+n (q p) . (4.1) U U t t t −

Since the empty set maximizes utility when the price is q , we get that (q )= (q )= 1 U 0 U 1 0. Moreover, the utility at price q =0is (q )=H (S). Thus we get that +1 U +1 v H (S)= n (q q ) . v t t − t+1 t=1 The following lemma is a slight variation of Lemma 8 of [8].

86 Lemma 4.1.3. Suppose a set S is being shown to a buyer B, with valuation function v, using a uniform price. Let H be any number such that H H (S). Let γ > 1, and let ≥ v p[t]=H /γt. Then, for any k 0, we have ≥ k (p[k]) 1 S H p[t] Φ(p[t]) U H (S) | | . | | ≥ γ 1 ≥ γ 1 v − γk t=1 − − Proof. Suppose that q >p[k] q , for some s l.Since (q )= (q ) = 0, and also s ≥ s+1 ≤ U 0 U 1 that q OPT,so 1 ≤

(p[k]) = (q )+( (p[k]) (q )) U U s U − U s s 1 − = ( (q ) (q )) + n (q p[k]) U t+1 − U t s s − t=1 s 1 − = n (q q )+n (q p[k]) . t t − t+1 s s − t=1 The above sum can be seen as an integral of the following step function f from p[k]to

q1: in the range [qt+1,qt), f takes the value nt. So we can upper bound it by an upper integral of f. Note that f(p) Φ(p) S , and also that f is a decreasing function. Thus ≤ | | ≤ | | we get

k 1 − (p[k]) Φ(p[t + 1]) (p[t] p[t + 1]) U ≤ | | − t=0 k 1 k − =(γ 1) Φ(p[t + 1]) p[t + 1] = (γ 1) Φ(p[t]) p[t] . − | | − | | t=0 t=1 Further, since (0) = H (S), and we have (p[0]) (p[k]) = p[k] f(x)dx S p[k], U v U − U 0 ≤ | | so (p[k]) H (S) S p[k]. U ≥ v − | |

Brieﬂy, Lemma 4.1.3 will be used as follows: if one of H ,H /2,H /4 ...H /2k is { } chosen uniformly at random and set as the uniform price for all items in S, then for a

suﬃciently large choice of k, the revenue obtained is Ω(Hv(S)/k). This will happen when

the right-hand-side of the equation in the lemma evaluates to Ω(Hv(S)). We shall frequently

use this lemma, and with H = Θ(H), our choice of k will be logarithmic in the number of items.

87 4.1.4 Optimizing with Unknown Parameters

Almost all our algorithms use the following lemma, which was implicitly mentioned in the Appendix of [8]. It tells us that strategies can be allowed to assume that it approximately knows the value of some parameters, as long as the parameters are not too large, since these assumptions can be removed by guessing the value of these parameters and getting it correct with inverse-polylogarithmic probability. The lemma below is applicable to the Item Pricing problem with multiple buyers, and to both static dynamic pricing strategies.

Lemma 4.1.4. Consider a pricing strategy that gives an α-approximation in expected S revenue, provided the seller knows the value of some parameter x to within a factor of 2. Then if the seller instead only knows that L x 0, there exists a pricing strategy that gives an O(α log x(log log x)1+) approximation S in expected revenue.

Proof. We construct a pricing strategy by approximately guessing the value of x,uptothe nearest power of 2, using a suitable distribution, at the beginning, and using this estimate in the given pricing strategy. Our revenue is assured only when our guess is correct, and we count only that revenue in our analysis. In the case where L and H are given, we guess x from the set L, 2L, 4L...H , so that { } our guess of x is correct within a factor of 2, with probability at least Ω(1/ log(H/L)). Since the given pricing strategy gives a revenue of Ω(OPT/α) in expectation when the guess is correct within a factor of 2, we get an expected revenue of at least Ω(OPT/α log(H/L)). In the second case, where the seller only knows that x 1, then we guess that ≥ i 1 1+ x =2 with probability ,wherec = ∞ 1/i log i,whichisﬁnite.Ifx is c(i log1+ i) i=1 between 2i and 2i+1, then the probability of guessing x correctly within a factor of 2 is at least Ω(1/i log1+ i)=Ω(1/(log x(log log x)1+)), so the expected revenue is at least Ω(OPT/(α log x(log log x)1+)).

88 4.2 Improved Lower Bounds for Static Uniform Pricing

In this section, we construct a 2Ω(√log n) lower bound for static uniform pricing strategies, improving upon the 2Ω(√log n). Note that this lower bound essentially matches the upper bound in [8]. Our lower bound holds even if the seller had full information and could choose the ordering among buyers. In our constructed instances, each buyer has a simple XOS function with only 3 additive components. We construct another instance where all buyer valuations are identical and XOS, though this common valuation function has more than 3 additive components. These instances demonstrate that the limitation of static uniform pricing strategies comes from the dependency introduced by limited supply of items, instead of coming from the complexity or heterogeneity of the buyer valuations. The following theorem summarizes our lower bound results about static uniform pricing.

Theorem 8. There exists a set of buyers with XOS valuations, such that if the seller is restricted to a static uniform pricing strategy, then even in the full information setting, for any choice of price, the revenue produced is at most OPT/2Ω(√log n), where n is the number of items. Additionally, one of the following (but not both) can also be ensured, with the revenue still being at most OPT/2Ω(√log n):

The valuations of all the buyers can be expressed as 3-XOS functions. • All buyers have identical valuation function. • We now present the proof of Theorem 8. We will ﬁrst construct an instance with two buyers whose valuations consist of only three additive components each, such that if buyer 1 arrives before buyer 2, then the revenue obtained will satisfy the required upper bound. This part of our construction is very similar to that given in [8], with some changes in the

1/4 parameter that allows us to improve the lower bound result from 2Ω(log n) to 2Ω(√log n). We will then extend this construction to instances with essentially the same revenue bound, where all buyers have identical valuations.

89 4.2.1 A Hard Two-Player Instance

Let us fix some parameters that we use in our construction: Let k be a (large) positive k 6k 6k2 2 integer, Y = √2, X = Y , and n0 = X = Y . The instance will have n = X + X + + X6k = Θ(n )items. ··· 0 n0 Let S0,S1, ,S6k 1 be a partition of the items into disjoint sets such that Si = i . ··· − | | X Si n For every i,lettherebeasubsetS S such that S = | | = 0 . i ⊆ i | i| X Xi+1 There are two buyers with valuations v1 and v2 respectively. Let us first define the valuation for each item. Then, we will complete the definition by giving the additive components. The buyer valuations for a single item are:

0,ifj S S ∈ i \ i v (j)= i 1  1 X  ,ifj S  n Y ∈ i 0  i  X X ,ifj Si Si 2n0(X 1) Y ∈ \ v2(j)= −  X X i  ,ifj S 2n Y ∈ i 0  The valuation function of each buyer consists of three additive components which are additive inside the subsets A0 = S0 S3 S6k 3, A1 = S1 S4 S6k 2, and ∪ ∪ ···∪ − ∪ ∪ ···∪ − A2 = S2 S5 S6k 1 respectively, that is, ∪ ∪ ···∪ −

v (S) = max v (j) , v (j) , v (j) , 1  1 1 1  j S A0 j S A1 j S A2  ∈∩ ∈∩ ∈∩    v (S) = max v (j) , v (j) , v (j) . 2  2 2 2  j S A0 j S A1 j S A2  ∈∩ ∈∩ ∈∩  .   We will prove that this instance admits the desired gap between the optimal social welfare and the revenue given by the best static uniform pricing strategy.

Lemma 4.2.1. In the two-player instance, the optimal social welfare is greater than 1,and no static uniform pricing strategies could give more than 1/2Ω(√log n) of revenue.

Proof. Let us ﬁrst consider the value of the optimal social welfare. By our construction, we have the following facts about buyer valuations.

90 Fact 1. For every 0 i 6k 1, we have ≤ ≤ − 1 1 1 v (S)= ,v(S)= ,v(S S)= . 1 i XY i 2 i 2Y i 2 i \ i 2Y i Fact 2. For each 0 j 2, we have ≤ ≤ 1 1 Y 6k 1 1 Y 6k v (A )= − − ,v(A )= − − . 1 i XY i 1 Y 3 2 i Y i 1 Y 3 − − − − By these facts, it is easy to see that allocating A1 to buyer 1 and A0 to buyer 2 maximizes the social welfare. Therefore, we have that in the two-player instance that we constructed, the optimal social welfare is 1 1 Y 6k 1 Y 6k OPT = v (A )+v (A )= − − + − − > 1 . 1 1 2 0 XY 1 Y 3 1 Y 3 − − − − Next, we will show that the revenue given by any static uniform pricing is at most O(OPT/X)=OPT/2Ω(√log n). Let us first briefly explain the intuition behind this proof. As we can see in the construction, the revenue that comes from buyer 1 is upper bounded 1 by her maximum valuation, which is only O X . So in order to get revenue non-trivially better than 1/X, we need to get good revenue from buyer 1. However, we will show that when the uniform price is chosen such that selling a subset of items, say, S S ..., i ∪ i+3 ∪ to buyer 2 would provide good revenue, buyer 1 would purchase the subset S S .... i ∪ i+3 ∪ 1 On the one hand, the size of the latter subset is only a X fraction of the size of the former. So the purchase of buyer 1 does not provides much revenue. On the other hand, the subset purchased by buyer 1 contribute half of the valuation in the subset S S .... i ∪ i+3 ∪ Therefore, buyer 2 now has much less incentive to purchase this subset and would turn to S S ... instead, which gives little revenue. i+1 ∪ i+4 ∪ Now let us present the proof which validates the above intuition. Let us illustrate which subset of items each buyer would choose given a uniform price p of the items. First, it is clear that the buyer would choose a subset of items from the same additive component. Second, given the price P and subject to that the buyer would choose items from the additive component A , the buyer would choose an item j A iff its valuation is higher i ∈ i than the price p. Notice the valuation for a single item in the same additive component is monotone in the sense that

v (j S)

S 6k i 1 for 0 i 6k 1. Similarly, for buyer 2 we only need to consider subsets i+3 −3 − ≤ ≤ − of the form S S S 6k i 1 and S S S S 6k i 1 for i i+3 i+3 − − i 1 i i+3 i+3 − − ∪ ∪ ···∪ 3 − ∪ ∪ ∪ ···∪ 3 0 i 6k 1. ≤ ≤ − For convenience, we let Ti denote j ζ Sj,whereζi = j :6k 1 j i, j i ∪ ∈ i { − ≥ ≥ ≡

(mod 3) . Similarly, we let Ti denote j ζi Sj .Weletui(S, p) denotes the utility of buyer } ∪ ∈ i at price p when she buys the set S. Notice that if p [v (j S S ),v (j S S )], then the total number of items ∈ 2 ∈ i \ i 2 ∈ i+1 \ i+1 n0 n0 which has valuation higher than p is at most Si + i+1 j<6k Sj = Xi+1 + i+1 j<6k Xj < | | ≤ | | ≤ 3n0 X X i+1 3n0 3 i+1 . So the revenue is at most i+1 < i+1 . Hence, we conclude that if X 2n0(X 1) Y X Y − 3k 3 OPT p>v2(j S3k 1 S3 k 1), then the revenue is at most O(Y − )=O(X− )=O √log n . ∈ − \ − 2 In the remaining discussion, we will assume p v2(j S3k 1 S3 k 1). ≤ ∈ − \ −

Choice of Items of Buyer 1 We have the following for every 0 i 3k 1: ≤ ≤ −

1 n0 u (T ,p)= v (S ) p S = p 1 i 1 j − | j| XY j − Xj+1 j ζi j ζi ∈ ∈ 1 3 n0 3 = (1 O(X− )) p (1 O(X− )) . XY i(1 Y 3) ± − Xi+1 ± − − Therefore, u (T ,p) u (T ,p) is equivalent to that the price p satisﬁes (0 i 1 i ≥ 1 i+1 ≤ ≤ 3k 1): − i 1 X 1 p a = (1 O(X− )) (4.2) ≤ i n (1 + Y 1 + Y 2) Y ± 0 − − Here, we omit the exact value of ai for the sake of clean notation. Notice that ai’s are

monotone, we have that buyer 1 would choose the subset Ti iﬀ. p [ai 1,ai]. The buyer ∈ − may break ties arbitrarily on the boundary.

Choice of Items for Buyer 2 Now we turn to buyer 2’s choice of items. Subject to the

assumption that p [ai 1,ai], 0 i 3k 1, the items in Ti are taken away by the ﬁrst ∈ − ≤ ≤ − i 1 buyer when buyer 2 arrives. Further, we have that v (j S S )= X X − < 2 i 1 i 1 2n0(X 1) Y ∈ − \ − −

92 ai 1. So the options available to buyer 2 are the following: − 1 n0(X 1) u (T T ,p)= v (S S ) p S S = p − 2 i \ i 2 j \ j − | j \ j| 2Y j − Xj+1 j ζi j ζi ∈ ∈ 1 3 n0 1 = (1 O(X− )) p (1 O(X− )) 2Y i(1 Y 3) ± − Xi ± − − 1 1 (1 O(X− )) (because p ai 1) , ≤ 2Y i(1 Y 3) ± ≥ − − − 1 n u (T ,p)= (v (S ) p S )= p 0 2 i+1 2 j − | j| Y j − Xj j ζi+1 j ζi+1 ∈ ∈ 1 3 n0 1 = (1 O(X− )) p (1 O(X− )) Y i+1(1 Y 3) ± − Xi+1 ± − − 1 1 (1 O(X− )) (because p a ) ≥ Y i+1(1 Y 3) ± ≤ i − − 1 1 u2(Ti+2 Si 1,p) v2(Sj)+v2(Si 1)= + ∪ − ≤ − Y j+2 2Y i 1 j ζ j ζ − ∈i+2 ∈i+2 1 3 1 = (1 O(X− )) + , Y i+2(1 Y 3) ± Y i+1 − − 1 3 u (T ,p) v (S )= (1 O(X− )) . 2 i+2 ≤ 2 j Y i+2(1 Y 3) ± j ζ − ∈i+2 − It is easy to verify that

u2(Ti+1,p) > max u2(Ti Ti,p),u2(Ti+2 Si 1,p),u2(Ti+2,p) . { \ ∪ − }

So subject to the assumption that p [ai 1,ai], we get that buyer 2 would choose the ∈ − subset Ti+1. Hence, the total revenue is at most

n0 n0 2 p( T + T ) a + < . | i | | i+1| ≤ i Xi+1 Xi+1 X Thus we get an Ω(X)=2Ω(√log n) gap between revenue and the optimal social welfare since log X = Θ(k)=Θ(√log n).

4.2.2 Extensions of the Two-Player Instance

We now complete the proof of Theorem 8 by extending the above two-player instance. We present three hard instances such that

Instance 1: The 2Ω(√log n) lower bound holds even if the buyers come in random • order;

93 Instance 2: The lower bound holds even if the seller can choose the order of the • buyers;

Instance 3: The lower bound still holds if all the buyers have identical valuation • functions.

The construction of each of the latter instances is based on the previous instance.

Instance 1: Consider an instance with n items, where there are m X buyers. One of ≥ them has the same valuation function as “buyer 2” in the two player instance constructed above. The other m 1 of them have identical valuation functions, and match that of − “buyer 1”. Each of the other m 1 buyers has its own shadow copy of T . Then the profit − i is at most OPT if the special buyer comes first and (following the arguments given for the OPT two-player instance above) at most O X otherwise. Since the first event happens with probability only 1/m, so the expected revenue is at most (since m X): ≥ 1 m 1 OPT OPT OPT + − O

Instance 2: Now consider another instance in which we make m copies of the setting in Instance 1 (only items are copied, not the buyers) such that each buyer is “buyer 2” in exactly one copy, that is, each buyer has the same valuation as buyer 2 of the original 2-player instance in exactly one copy, and is “buyer 1” in the remaining copies. Moreover, each copy of the set of items has exactly one buyer designated as “buyer 2”. We combine these valuations over the distinct sets of items to get 3-XOS valuations: the valuations of the

sets Ai of each copy, for each i =0, 1, 2, are combined in an additive fashion to create each

additive component. Let OPT1 denote the optimal social welfare in Instance 1. Then, it is

easy to see that OPT = Ω(mOPT1). When the ﬁrst buyer in this new instance comes, she

may provide “good” revenue (at most OPT1) by purchasing items in the replicate in which she is buyer 2. However, she also ruins all other replicates since she behaves as “buyer 2” in those replicates. Therefore, by our construction, each of the remaining buyers will provide

OPT1 at most O X of revenue. This happens regardless of who comes in as the ﬁrst buyer. So the total revenue is at most OPT +(m 1)O OPT1

94 Instance 3: Finally, consider a setting where all buyers are identical and share the same valuation function v(S) = max1 i m vi(S), where vi 1 i m are the buyer valuation ≤ ≤ { | ≤ ≤ } functions as deﬁned in Instance 2. This is no longer a 3-XOS valuation. Intuitively, each of these newly constructed buyers can buy items from any one copy of the item set behaving as “buyer 2”, but if it does so, then it must act as “buyer 1” in all the other copies. So as in Instance 2, the ﬁrst buyer to arrive buys St in one of the copies and S in the all t other copies for some t, t, and ruins the revenue obtainable on all the other copies from the remaining buyers when it does so. Therefore, a static uniform pricing strategy has the same performance here as in Instance 2. This completes the proof of Theorem 8.

4.3 Dynamic Uniform Pricing Strategies

We now present a dynamic uniform pricing strategy that achieves an O(log2 n)-approximation to the revenue when buyer valuations are subadditive. This improves upon the previous best known approximation factor of 2O(√log n log log n) [8] for the Item Pricing problem. Our strategy makes the assumption that the seller knows OPT, the maximum social welfare, to within a constant factor. However, this assumption can easily be eliminated by using Lemma 4.1.4, worsening the approximation ratio of the strategy by a poly-logarithmic factor. As noted earlier, our algorithmic results for uniform strategies extend to the setting where items may have multiple copies, by treating each copy as a distinct item. We will also establish an almost matching lower bound result which shows that no dynamic uniform pricing strategy can achieve o(log2 n/ log log2 n)-approximation even when buyers are restricted to XOS valuations, the seller knows the value of OPT, buyer valuation functions, and is allowed to specify the order of arrival of the buyers.

4.3.1 A Dynamic Uniform Pricing Algorithm

Let k = log n + 1, and let p = OPT/2i (recall that OPT denotes the maximum social i welfare). The pricing strategy proposed by Blum et. al. [8] is to simply choose a price p from p ,1 i k, uniformly at random; then set up a static uniform strategy with price p. i ≤ ≤

95 Even though this strategy has poor performance when there are two buyers and a limited supply of items, Blum et. al. showed that if there is only a single buyer (equivalently, there are unlimited number of copies available for each item) the expected revenue of this strategy is an O(log n) approximation to optimal social welfare. Our lower bound construction on static uniform pricing with two buyers would in fact break if the two buyers could be shown two different (but uniform) prices. So it is natural to consider the straightforward dynamic extension of the simple strategy in [8], namely, pick a fresh uniform price p at random upon the arrival of each buyer. However, the following instance indicates that such a pricing strategy may perform poorly: consider 2n buyer and n items; the first n buyers share an additive valuation and have value 1 for each item; the last n buyers are unit-demand buyers that have value n for getting at least item (getting more than one items does now increase the valuation). It is easy to verify that OPT = n2 because we can allocate one item to each of the last n buyers. However, if we use the “fresh-price-upon-arrival” strategy as described above, then with high probability, we would have choose a uniform price that is at most 1 for at least one of the first n buyers; hence the buyer would have purchase all the items at a very low price. Now, let us introduce a dynamic uniform pricing strategy which avoid selling items at low price by picking a non-trivial reserve price upfront:

1: At time 0, choose a reserve price p from the set p ,p ...p , uniformly ∗ { 1 2 k+1} at random.

2: Upon arrival of any buyer, the algorithm chooses a pricep ˆ uniformly at random from the set p ,p ...,p , and assigns the pricep ˆ to all items that are yet { 1 2 ∗} unsold.

Let us ﬁrst give a short argument which shows that this strategy gives a revenue of Ω(OPT/ log3 n) in expectation. This argument shall provide some intuition why the above pricing strategy guarantees good revenue.

Consider a social welfare maximizing allocation (T1,...,Tm). By the standard argument in [43] and [8], there exists a price p p ,...,p and subsets T T supported at p such ∈ { 1 k} i ⊆ i that p m T = Ω OPT . Further, if the algorithm guesses the correct reserve price, i=1 | i| log n that is,p∗ = p/2, then upon the arrival of buyer i we have the following holds: Either

96 half of the items in Ti are already sold, in which case take into account the revenue of at least p ( T /2) = p T /4 for selling those items in T ; or at least half of the items in T ∗ | i| | i| i i remain unsold, in which case we will take into account the revenue we get from buyer i. Subadditivity ensures that the unsold set of items are still valuable, regardless of the items that have been sold. Due to an analogue of Lemma 4.1.3 (see Lemma 4.3.1 for details), picking a random uniform price from p ,p ,...,p provides an expected revenue of at { 1 2 ∗} least Ω(p T / log n) from buyer i. In the above argument, we are counting the revenue from | i| each item at most twice (for the set Ti it belongs to; and for the buyer who buy it). So the expected revenue when p = p/2 is at least n Ω(p T / log n)=Ω(OPT/ log2 n). Since ∗ i=1 | i| p∗ = p/2 with probability Ω(1/ log n), we have the desired bound. Next, we will provide a more careful analysis of the algorithm by taking into account the revenue contribution when the algorithm makes a “wrong” choice of reserve price. By doing so, we are able to shave oﬀ a log n factor and prove that the expected revenue is at least Ω(OPT/ log2 n).

Theorem 9. If the buyer valuations are subadditive, then the expected revenue obtained by the dynamic strategy above is Ω(OPT/ log2 n).

The following lemma is key to the proof of Theorem 9. It gives a lower bound on the expected revenue obtained from a buyer if there remains a subset of items that is supported at twice the reserve price.

th j Lemma 4.3.1. Suppose when the i buyer Bi arrives, there remains a set Li of unsold items such that v (Lj) p Lj , where v is the valuation function of B . Then if the seller i i ≥ j| i | i i picks a price from p ,p , ,p uniformly at random, and prices all items at this single { 1 2 ··· j+1} price, it receives an expected revenue of at least p Lj /2(j + 1) from this buyer. j| i |

Proof. (Lemma 4.3.1) Let I be the set of unsold items when the buyer Bi arrives. Now if j the uniform price chosen by the seller is pj+1,thenbuyingthesetLi would give Bi a utility of at least v (Lj) p Lj p Lj p Lj since v (Lj) p Lj by the assumption of i i − j+1| i | ≥ j| i | − j+1| i | i i ≥ j| i | the lemma. Thus j j j pj Li (B ,I,p ) p L p L = | | (4.3) U i j+1 ≥ j| i | − j+1| i | 2

97 From Lemma 4.1.3, we get that (B ,I ,p ) j+1 Φ(B ,I ,p ) p . Combining with U i j+1 ≤ t=1 | i t | t equation 4.3, we get j+1 j pj Li Φ(B ,I,p ) p | | . | i t | t ≥ 2 t=1 Thus the expected revenue obtained from Bi is

j+1 j 1 pj Li Φ(B ,I,p ) p | | , j +1 | i t | t ≥ 2(j + 1) t=1 completing the proof of the lemma.

Proof. (Theorem 9) Let (T1,T2,...Tm) be an optimal allocation of items to buyers B1,B2 ...Bm, m who have valuation functions v1,v2 ...vm respectively, such that i=1 vi(Ti)=OPT is the j maximum social welfare. Also, let Ti be the subset of Ti that would be bought by Bi if it

were shown only the items in Ti, and all items were uniformly priced at pj. Now consider the case when p = p . Let Rj be the revenue in this case. Let Zj T j be a random ∗ j+1 i ⊆ i j variable that denotes the subset of items in Ti that are sold before buyer Bi comes. Then Rj m p Zj = m p Zj /2. ≥ i=1 ∗| i | i=1 j| i | j j j j Note that vi(T Z ) pj T Z by Lemma 4.1.1. So, by Lemma 4.3.1, conditioned i \ i ≥ | i \ i | on the set Zj, the expected revenue received from B is at least p T j Zj /2(j + 1). i i j| i \ i | j Thus, conditioned on the sets Zi for all i,wehave

m j j m j j j j pj Ti Zi pj Ti E[R Zi 1 i m] Ω pj Zi + | \ | = Ω | | . | ∀ ≤ ≤ ≥ | | j j i=1 i=1 j Since the value on the right-hand side above is independent of the variables Zi on which the expectation of Rj is conditioned on, we get

m p Tj E[Rj]=Ω j| i| . j i=1 k j Thus the expected revenue R = j=0 R of our dynamic strategy is given by

k k m m k 1 p Tj p Tj E[R] = E[Rj]=Ω j| i| = Ω j| i| (4.4) k+1  k2   k2  j=0 j=0 i=1 i=1 j=0    

98 Since k = log n + 1, and OPT H (T ), from Lemma 4.1.3 and Equation (4.4), it ≥ vi i follows that k T OPT p T j Ω v (T ) | i| . j| i | ≥ i i − 2n j=0 Thus we have

m m 1 Ti OPT E[R] = Ω 2 vi(Ti) | | k − 2n i=1 i=1 1 OPT OPT = Ω OPT = Ω . k2 − 2 log2 n

4.3.2 Lower Bound for Dynamic Uniform Pricing

We shall now construct a family of instances of the problem, where the buyers have distinct -XOS valuations, where = O(log n/ log log n), such that no dynamic uniform strategy can achieve a o(log2 n/(log log n)2)-approximation, even in the full information setting, and even when the seller can specify the order in which the buyers should arrive. Before getting into the full proof, let us ﬁrst explain the high-level picture of our hard instance. First, let us review two instances of a single buyer which yield an Ω(log n/ log log n) lower bound for uniform pricing. We will use these two instances as basic gadgets in the lower bound construction.

Instance 1: Consider k = Θ(log n/ log log n) disjoint subsets of items S1,...,Sk such that S = n/ log2i n and v(S )=i for i [k]. Here we let v denote the | i| i ∈ buyer’s valuation. v is an XOS valuation with k additive components that are additive among items in S , i [k], respectively. i ∈ Instance 2: Almost the same as instance 1 except that the valuation v is additive among all items.

In instance 1, we have that OPT = k and the best revenue by uniform pricing is O(1). In instance 2, we have that OPT = Ω(k2) and the best revenue by uniform pricing is O(k). The analysis is deferred to the full proof. Hence, each of the two instances establishes an Ω(k)=Ω(log n/ log log n) lower bound.

99 Now let us explain how to construct a hard instance based on these two gadgets. We will consider m log n buyers. Each buyer has a private copy of instance 1 and a public copy of instance 2. Here, the terms private and public are used to indicates if the other buyers are interested in buying those items. Concisely, each buyer does not value the items in the other buyers’ private subset, but do value the items in the other buyers’ public subsets, yet at a much lower value which is only a 1/k2 fraction of value of the same items in her own public subset. On the one hand, allocating each agent her public subset yields a social welfare of Ω(mk2). So we have OPT = Ω(mk2). On the other hand, by our construction, if a buyer purchase items in her private subset, than the revenue is no more than O(1). So in order to get revenue at least ω(m), we need to have some “good” sale by selling items in the public subsets. Indeed, the buyers may have incentive to choose the public subsets because the valuation in instance 2 is additive among all items. However, we can further show that when a buyer buys items in her public subset, she also buys many items from other buyers’ public subsets at low price. Therefore, she eliminates (partially) the edge of the public subsets. Moreover, after a few such “good” sales (at most O(log n)), the advantage of buying the public subsets has completely disappeared. Hence, we can bound the number of such “good” sales by O(log n) and further bound the total revenue from these “good” sales, getting the desired lower bound. Next, let us formally state our lower bound result in Theorem 10 and present the proof.

Theorem 10. There exists a set of buyers with XOS valuations, such that if the seller is restricted to using a dynamic uniform pricing strategy, then even when the seller has full information of buyer valuation functions and can even choose the order of arrival of the buyers, the revenue produced is O(OPT(log log n)2/ log2 n), where n is the number of items.

Proof. Let B1,B2 ...Bm denote the buyers. Our construction will use three integer parameters k, F, and Y , to be speciﬁed later. These parameters will satisfy the conditions that k>1, F>1, Y>4, and m 2Y 4k. Let f(i)=(i + 1)F/Y i.Then, ≥ ≥ f(0) >f(1) >...>f(k) >f(k + 1).

For each buyer Bi, we create 2(k+1) disjoint sets of items Si0,Si1 ...Sik and Si0,Si1 ...Sik j such that Sij = Sij = Y items each. Let Si = 0 j kSij and Si = 0 j kSij . We call | | | | ∪ ≤ ≤ ∪ ≤ ≤

100 the items in Si as shared and those in Si as private. The private items of Bi are valued by buyer i only, and has zero value to all other buyers.

The valuation function vi of buyer Bi is constructed as an XOS valuation with (k + 2) additive functions vi0,vi1 ...vi(k+1) in its support, that is, vi = max0 j k+1 vij. For 0 ≤ ≤ ≤ j k, the valuation function v has positive value only for private items, and is deﬁned as ≤ ij

f(j)ifx Sij vij(x)= ∈  0 otherwise 

The valuation function vi(k+1) has positive values only for shared items:

f(j)ifx Sij for 0 j k vi(k+1)(x)= ∈ ≤ ≤  f(j + 1) if x S for 1 m, = i, and 0 j k  ∈ j ≤ ≤ ≤ ≤

This completes the description of the instance. Note that vi(S )=f(j) S =(j +1)F , ij | ij| and that k k v (S )= f(j) S = (j + 1)F = Ω(k2F ) . i i | ij| j=0 j=0 Thus if we allocate each set Si to buyer Bi for i =1, 2 ...m, the social welfare obtained is Ω(mk2F ), and hence OPT is Ω(mk2F ).

Consider now the arrival of some buyer Bi at time t. By our construction of the valuation function, Bi will either buy only shared items or buy only private items, but not both. If the buyer B were to buy shared items, and the price of each item is set at f(j) p>f(j + 1), i ≥ then Bi would pick up all remaining items in

S S .  it  t 0 t j 1 =i m 0 t (j 1) ≤≤ ≤ ≤ ≤ ≤− t j    Since 0 t j Y 2Y , the total price that Bi would pay to the seller is bounded by ≤ ≤ ≤ j j 1 f(j)(2Y +2mY − ) = 2(j + 1)F +2m(j + 1)F/Y = 2(1 + m/Y )(j + 1)F.

We now consider the maximum revenue generated if Bi were to buy a subset of its private items. Note that when Bi arrives, all private items of Bi are still unsold. Suppose

Bi were to buy private items. What is the maximum revenue we can get? For this, note j that if the price of each item is (j + 1)F/jY , then the utility from buying Sij is

(j + 1)F (j + 1)F/j =(j + 1)(j 1)F/j =(j 1/j)F, − − − 101 and the utility from buying Si(j 1) is −

jF (j + 1)F/jY > (j 1/j2)F>(j 1/j)F ,sinceY>2j. − − −

So at this price, the set Si(j 1) is preferred to Sij by Bi, and since the items in sets Sit − for t>jhave less value than the price, they are not even considered. For a greater price,

the utility of Si(j 1) must continue to dominate that of Sij , since the former has fewer items. − So at most Y j 1 items are bought when the price is at least (j + 1)F/jY j, for all j 1. − ≥ This implies that the revenue obtained from Bi when she buys from her private items is at most Y j((j + 1)F/jY j) < 2F . Consider any ordering of buyers. If the price is ever set at more than f(0), then no item is sold in that round, while if the price set is f(k + 1) or lower, all items are sold in that round and the revenue generated is at most 2mY kf(k + 1) = 2m(k + 2)/Y . Consider the ﬁrst time when the price set in a round is at most f(j) but greater than f(j + 1), for some 0 j k. We call this round a j-good sale, and let B be the buyer. In a j-good ≤ ≤ i sale, B may buy all remaining items in S for all 0 t j, plus all items in S for all i it ≤ ≤ lt 1 m, = i and 0 t j 1, to yield revenue of at most ≤ ≤ ≤ ≤ −

2(1 + m/Y )(j + 1)F O((m/Y )kF) . ≤

However, consider any time when a price in the range (f(j 1),f(j)] appears again, and − let B , = i be the buyer who faces this price. If B were to buy shared items, the only l l items that are valued higher than the price and still remaining are those in Sj,sinceBi

took away whatever was remaining of St for all t

items could have given Bi a better utility was that the shared items had additive valuation, while the private items had XOS valuation, so she got no beneﬁt in picking up multiple sets

of private items. However, since only one feasible set Sj of the shared items is left, this

advantage has vanished, and the revenue from B is the same as the revenue if there were

no shared items at all. As discussed above, the revenue from B in this case is at most 2F . Finally, since a j-good sale can happen at most once for any 1 j (k + 1), the total ≤ ≤ revenue generated fro all j-good sales is O (mk2F )/Y . The remaining rounds each give a revenue of at most 2F , contributing in total O(mF ) to the revenue. Thus the revenue obtained by any dynamic uniform strategy, for any ordering of buyers,isO (1 + k2/Y )mF . 102 Now since the maximum social welfare is Ω(k2mF ), the approximation factor achieved is bounded from below by Ω (k2Y )/(k2 + Y ) . For any k>10, if we set Y = k2 and m =2Y ,thenn = Θ(Y k+1)= kΘ(k), and the approximation factor is Ω(k2). As k = Θ(log n/ log log n), we get that the smallest approximation factor that can be achieved is Ω (log n/ log log n)2 .

4.3.3 Dynamic Monotone Uniform Pricing Strategies

The dynamic uniform strategy described in Section 4.3.1 chooses a random price for each buyer; this can result in large ﬂuctuations in prices shown even to consecutive buyers, which may not be preferable. We now present a simple strategy that uses a monotonically decreasing uniform pricing for the items. When the number of buyers m is at least 2 log n, the strategy gives an O(log2 n)-approximation to the revenue provided the buyers arrive in a uniformly random order, that is, all permutations of the buyers are equally likely to be the arrival order. As a corollary of this result, we conclude that if the buyers are identical, no matter the order in which they arrive, this pricing scheme gives an O(log2 n)-approximation. The strategy assumes that the seller knows the number of buyers m (and also OPT), and is deterministic. Knowing estimates of m and OPT up to constant factors are also suﬃcient for the performance of our strategy. Just like OPT, an estimate for m can be guessed using Lemma 4.1.4. k m Let k = log n + 1, and let γ =2m 1. Thus γ > 2n. The strategy gives a good ≥ guarantee only when m log n + 1. The strategy is as follows: ≥ 1: For 1 t m, the seller prices all unsold items uniformly at ≤ ≤ OPT p[t]= 2γt

when the tth buyer arrives.

Thus the price decreases with time. For m = ω(log n), the relative decrease in the price for consecutive buyers is

p[t] p[t + 1] 1 log n − = 1 = Θ( ) , p[t] − γ m

103 which tends to zero, and so the price decreases smoothly with time. It is worth noting that another dynamic monotone strategy yields the same approximation guarantee (similar m analysis, details omitted): start with price OPT, and halve the price after every log n buyers. This strategy has the beneﬁt that the price is changed only a few (log n)times.

Theorem 11. Suppose m log n+1, and suppose that the buyer valuations are subadditive. ≥ If the ordering of buyers in which they arrive is uniformly random (that is, all permutations are equally likely), then the expected revenue of the dynamic monotone uniform pricing scheme described above is OPT Ω . log2 n Proof. The proof of this theorem is fairly complex, so let us ﬁrst provide some intuition

regarding its veracity. If we focus on the contribution of a single buyer Bi after ﬁxing the

sequence of the remaining buyers (Bi is inserted at a random position in this ﬁxed sequence), then the following two properties hold, as they did for our ﬁrst dynamic (non-monotone)

uniform strategy: Bi is oﬀered a random price from a geometric range of prices, and all previously sold items have been sold at a signiﬁcantly high price. This observation invites

a similar analysis: if a large number of items from the target set of Bi were sold prior to her arrival, that would contribute a large fraction of the revenue that could be obtained

from buyer Bi; else a random price shown to Bi yields good revenue from the remaining items. The second part of the analysis required that the random price oﬀered is independent from the set of remaining items, which unfortunately is not true for our dynamic monotone

strategy: the position of Bi in the sequence aﬀects both the price oﬀered as well as the set of remaining items. So we need a more intricate analysis – one that does not merely conditions on the set of remaining items. Nonetheless, our intuitions are proved correct, as we see below. With this prelude, we present our formal proof of the result.

Let (T1,T2,...Tm) be an optimal allocation of items to buyers B1,...,Bm, who has m valuation functions v1,v2 ...vm respectively, such that i=1 vi(Ti)=OPT is the maximum j social welfare. Also, let Ti be the subset of Ti that would be bought by Bi if it were shown j only the items in Ti, and all items were uniformly priced at OPT/γ =2p[j].

Fix a buyer Bi. Let Ri be a random variable that denotes the revenue obtained by

the seller from Bi. Let Ri be a random variable that denotes the revenue obtained by the

104 seller by selling items in Ti.Then,ifR is a random variable that denotes the total revenue R +R obtained by our strategy, we have R = m R and R m R ,soR m i i . i=1 i ≥ i=1 i ≥ i=1 2 Fix a permutation π of all buyers except Bi. We shall say that the event π occurs if these buyers arrive in the relative order given by π,withBi arriving somewhere in between. We shall now compute E[R +R π]. i i| Let π denote the permutation of all the buyers formed by inserting B after the (j 1)th j i − but before the jth position in π, whichever exists, for 1 j m. That is B comes in as ≤ ≤ i th j the j buyer in πj. Let Zi denote the set of items that were sold before the arrival of Bi j when the arrival sequence of buyers is πj. Note that Zi is no longer a random variable once π is ﬁxed, and neither are R and R . Also note that Pr [π π]=1/m. Thus, j i i j| m m 1 j 1 j E[R π] p[j 1] Z p[j] Z (4.5) i| ≥ m − | i| ≥ m | i| j=1 j=1 Let Sj be the set of items bought by B when the permutation of buyers is π . Let j i i j Ui j be the utility derived by Bi in this process, and let Ri be the revenue obtained from Bi in the process. For 1 j

j = v (Sj) p[j] Sj v (Sj+1) p[j] Sj+1 Ui i i − | i | ≥ i i − | i | = v (Sj+1) p[j + 1] Sj+1 (γ 1)p[j + 1] Sj+1 = j+1 (γ 1)Rj+1 i i − | i | − − | i | Ui − − i

j+1 1 j+1 j 1 This implies that Ri γ 1 ( i i ), for 1 j

j j t 1 j j 1 1 1 j R − + = i ≥ γ 1 Ui − Ui Ui γ 1Ui t=1 t=2 − − By Lemma 4.1.1, we have v (T j Zj) 2p[j] T j Zj . So the utility of T j Zj to buyer i i \ i ≥ | i \ i | i \ i B at price p[j]is(2p[j] p[j]) T j Zj = p[j] T j Zj , which is at most j.Thus i − | i \ i | | i \ i | Ui j 1 p[j] T j Zj Rt j = | i \ i | i ≥ γ 1Ui γ 1 t=1 − −

105 Using the above equation, we get

m m j m 1 m Rj Rt = p[j] T j Zj i ≥ i γ 1 | i \ i | t=1 j=1 j=1 − j=1 m m j j j j j p[j] Ti Zi p[j] Ti Zi Ri | \ | Ω | \ | ⇒ ≥ (γ 1)m ≥ log n j=1 j=1 − The last inequality follows from the fact that γ 1=Θ log n . − m Note that E[R π]= 1 m Rj. Combining with equation (4.5), we get that i| m j=1 i m j j 1 Ti Zi j E[Ri +Ri π]= Ω p[j] | \ | + Zi | m  log n | | j=1  m  m 1 p[j] T j 1 OPT Ω | i | = Ω T j ≥ m  log n  m log n  γj | i | j=1 j=1      Using Lemma 4.1.3, we get that

m OPT 1 T OPT 1 T OPT T j v (T ) | i| v (T ) | i| γj | i | ≥ γ 1 i i − γm ≥ γ 1 i i − 2n j=1 − − Again using the fact that γ 1=Θ log n , we get that − m 1 Ti OPT E[Ri +R π] Ω vi(Ti) | | i| ≥ log2 n − 2n Since the right-hand-side of the above equation is independent of π, we conclude

1 Ti OPT E[Ri +R] Ω vi(Ti) | | . i ≥ log2 n − 2n Thus we get that the expected revenue is

m m 1 1 Ti OPT E[R] = E[Ri +Ri] Ω 2 vi(Ti) | | 2 ≥ log n − 2n i=1 i=1 1 OPT OPT = Ω OPT = Ω log2 n − 2 log2 n

106 4.4 Static Non-Uniform Pricing

Another approach to get around the performance barrier for static uniform pricing, exhibited by Theorem 8, is to consider static non-uniform pricing, which allows the seller to post different prices for different items but the prices remain unchanged over time. We showed that there exist instances with identical buyers where no static uniform pricing can achieve better than 2Ω(√log n)-approximation even in the full information setting. Surprisingly, this hardness result breaks down if we consider non-uniform pricing, using only two distinct prices. As mentioned earlier, our algorithmic results extend to the setting where there are multiple copies of an item; however, our non-uniform strategies may assign different prices to different copies of the same item.

4.4.1 Full Information Setting

We first introduce the (p, )-strategies, i.e. the seller posts price p for a subset of the ∞ items and posts for all other items. The intuition is that by using this strategy the ∞ seller can prevent certain buyers from buying certain items (that has high utility to some other buyer) and thus achieve better revenue. The proof of the theorem below depends on the performance of the following dynamic monotone strategy. Let k = log n + 1 and m = m/(k + 1) . Recall that p = OPT/2i for i =1, 2, ,k. The seller posts a single i ··· price p1 for the first m buyers, then she posts a single price p2 for the next m buyers, and so on and so forth. We call each time period that the seller posts a fixed price a phase, and we call this strategy the k-phase monotone uniform strategy. The proof of Theorem 11 can be easily modified to show that this strategy gives O(log2 n)-approximation as well.

Theorem 12. In the full information setting, if m log n+1, and all buyers share the same ≥ subadditive valuation function, then there exists a (p, )-strategy which obtains revenue at ∞ least Ω(OPT/ log3 n).

Proof. Given that the k-phase dynamic monotone uniform strategy for identical buyers obtains revenue at least Ω(OPT/ log2 n), at least one of the k = log n +1 phases contributes 1/k fraction of the revenue. Without loss of generality, assume the ith phase contributes at least Ω(OPT/k log2 n)=Ω(OPT/ log3 n) revenue. Suppose T is the set of items unsold at

107 the beginning of phase i in the above k-phase strategy, and this set can be computed in the full information setting by simulating buyer behavior using oracle queries. Consider the following (p, )-strategy. The seller posts price p = p for each item in ∞ i+1 T , and posts for all other items. Then when the ﬁrst m = m/(k + 1) buyers come, ∞ they will behave the same as the m buyers in phase i in the dynamic strategy scenario. So the revenue collected is at least Ω(OPT/ log3 n).

4.4.2 Buyers with -XOS Valuations

The above theorem shows a clear gap between the power of uniform pricing and the power of non-uniform pricing in the full information setting. However, it crucially uses the knowledge of the valuation function and the fact that all buyers are identical; information that is usually not known to the seller. Hence strategies in the limited information setting are more desirable in practice. Fortunately, we find that considering static non-uniform pricing is also beneficial in the limited information setting. We first note that if the buyer order is randomized, then it is quite easy to get an O m log n log OPT(log log OPT)2 approximation using static uniform pricing, even with general valuations, and without the assumption of knowing OPT.This

can be done as follows: Just focus on selling items to the ﬁrst buyer. If Bi is the ﬁrst

buyer, and the algorithm knew the value vi(Ti), then using the single buyer (unlimited

supply setting) algorithm in [8], the strategy gets Ω(vi(Ti)/ log n) in expectation from the ﬁrst buyer, and we do not care what it gets from the other buyers. Thus the expected m 1 i=1 vi(Ti) OPT revenue of the algorithm is m log n = m log n . This algorithm would have to guess v (T ) OPT of the ﬁrst buyerB , up to a constant factor, and can do so by incurring an i i ≤ i additional factor of O(log OPT(log log OPT)2) as described in Lemma 4.1.4. However, if we require a strategy to give guarantees on expected revenue against any order of buyers, and in particular an adversarial ordering, then static uniform pricing cannot give a better bound than 2Ω(√log n) even when there are only two buyers, with 3-XOS valuations. This is evident from the proof of Theorem 8. We now show a static non- uniform strategy which achieves polylogarithmic approximation if we assume the valuation functions are -XOS functions where is quasi-polynomial in n and the number of buyers

108 is polylogarithmic, for all ordering of buyers. Let k = 2 log n , and let p = OPT/2i. With probability half, the seller assigns a single i price p randomly drawn from p ,p , ,p to all items. With probability half, the seller { 1 2 ··· k} assigns one of p ,p , ,p uniformly at random for each item. The price assignment 1 2 ··· k+1 remains unchanged over time.

Theorem 13. For m buyers with -XOS valuations functions, the expected revenue of the above strategy is OPT Ω . m log log3 n th Suppose the XOS valuation function of the i buyer is vi(S) = max1 j ai,j(S), where ≤ ≤ a (S) are additive functions i, j. For each m-tuple z =(z ,z , ,z ) []m,deﬁnea i,j ∀ 1 2 ··· m ∈ z to be an additive function such that for each item g I, az(g) = max1 i m ai,z (g). For ∈ ≤ ≤ i each z and each 1 i k,letΓz,j denote the set of items g such az(g) [pj,pj 1). We ≤ ≤ ∈ − say such a set Γz,j is large if its size is at least 16m log and we say it is small otherwise.

Deﬁne Az and Cz as follows:

Az = Γz,j ,Cz = Γz,j .

Γz,j 16m log Γz,j<16m log 1≥j k 1 j k ≤ ≤ ≤ ≤ In the case where the seller posts one of p ,p , ,p uniformly at random for each 1 2 ··· k+1 item, let Πj denote the set of items which are priced pj/2=pj+1. The following two lemmas are crucial to the proof of Theorem 13.

Lemma 4.4.1. If the seller posts a single price p randomly drawn from p , ,p for all { 1 ··· k} items, then the expected revenue is at least Ω(a (C )/m log log n) for any z []m. z z ∈

Proof. Let Rj denote the revenue if the seller posts a single price pj. When the seller posts a single price p for all items, the buyers will buy at least one item if C Γ is not empty. i z ∩ z,j Since C Γ < 16m log ,wehaveR a (C Γ )/16m log .Sincek = 2 log n , | z ∩ z,j| j ≥ z z ∩ z,j the expected revenue is at least

k k 1 a (C Γ ) a (C ) a (C ) R z z ∩ z,j = z z = Ω z z , k j ≥ 16km log 16km log m log log n j=1 j=1

since with probability 1/k, the price posted is pj and the revenue is Rj.

109 Lemma 4.4.2. If the seller posts one of p ,p , ,p uniformly at random for each item, 1 2 ··· k+1 then with probability at least 3/4 we have for every z []m and 1 j k, Π Γ ∈ ≤ ≤ | j ∩ z,j| ≥ A Γ /2k. | z ∩ z,j| Proof. If Γ is small then A Γ = 0 and the given equation is trivially true. Now z,j | z ∩ z,j| suppose Γ is large, that is, Γ 16m log . Note that each item in Γ has probability z,j | z,j| ≥ z,j 1/k of being priced pj/2. Using Chernoff bounds and we get that the probability that less 2m log 2m than 1/2k fraction of Γz,j are priced pj/2 is at most 1/2 =1/ . There are at most m distinct m-tuples z. For each z there are at most k = 2 log n sets Γ . So the total z,j m 2m number of different Γz,j is at most k< /4. By using union bound we finish the proof of this lemma.

We can now complete the proof of Theorem 13.

Proof. (Theorem 13) If there exists some vector z such that a (C ) OPT ,thenwe z z ≥ 320 log2 n know from Lemma 4.4.1 that the expected revenue is at least

OPT Ω . m log log n Now let us assume a (C ) < OPT for any z. z z 320 log2 n By Lemma 4.4.2, it suﬃces to prove that the expected revenue is high if for each z []m ∈ such that A Γ Π Γ | z ∩ z,j| . | j ∩ z,j| ≥ 2k Suppose T =(T ,T , ,T ) is the allocation that maximizes the social welfare, then 1 2 ··· m m m OPT = vi(Ti). There exists m-tuple z [] such that ai,z (Ti)=vi(Ti) and thus i=1 ∈ i m OPT = ai,z (Ti) az (I)=az (Az )+az (Cz ) . i ≤ i=1 By our assumption a (A ) OPT OPT/320 log2 n OPT/2 and hence a (A z z ≥ − ≥ z z ∩ Γ ) OPT/2k for some r [k]. Let Z denote the set Π Γ and we have Z z,r ≥ ∈ r ∩ z,r | | ≥ Γ /2k.Sincek = 2 log n ,wehave z,r pr Γz,r az (Γz,r) OPT OPT pr Z 2 2 . | | ≥ 2k ≥ 4k ≥ 8k ≥ 40 log n 110 Suppose the ith buyer buys the set S for 1 i m and let S denote the union of i ≤ ≤ all S .IfS Z Z /2 then the revenue is at least (p /2) S Z (p /2)( Z /2) = i | ∩ | ≥ | | r | ∩ | ≥ r | | Ω(OPT/ log2 n). Otherwise, Z S Z /2. Let u (S ) denote the utility of set S to the | \ | ≥ | | i i i ith buyer. We have

m m pr pr OPT ui(Si) ui(Z S) az (Z S) Z S Z S . ≥ \ ≥ \ − 2 | \ | ≥ 2 | \ | ≥ 160 log2 n i=1 i=1 Hence m v (S ) m u (S ) OPT . Note that there exists an m-tuple z [l]m i=1 i i ≥ i=1 i i ≥ 160 log2 n ∈ such that a (S )=v (S ). So i,zi i i i

m OPT az (Cz )+az (Az )=az (I) ai,z (Si) . ≥ i ≥ 160 log2 n i=1 By our assumption, a (C ) < OPT ,soa (A )=Ω(OPT/ log2 n). Note that an item z z 320 log2 n z z

g is bought if and only if its price is less than ai,z (g) for some i. So all items in Γz ,j Πj i ∩ are bought (for all j) and by Lemma 4.4.2, with probability at least 3/4, the revenue is at least

k k p Γ A pj j z,j z az (Az ) OPT Γz ,j Πj ∩ = Ω = Ω . 2 ∩ ≥ 4k k log3 n j=1 j=1 Hence the expected revenue is at least Ω(OPT/m log log3 n), and the proof is complete.

4.5 Conclusion

We have shown that using multiple prices allows a seller to get better guarantees on revenue even when the seller does not know the buyers’ valuations. For this, we allowed the seller to show diﬀerent prices to each buyer. These dynamic strategies, however, still show the same price on every item to a particular buyer. Dynamic strategies may be applicable in may settings, particularly online settings, but static strategies are more widely applicable. We have also shown that static strategies using multiple prices can be better than using a single price on all items, but our guarantee is better only when the number of buyers is small, and the

111 valuation functions are more restrictive than just being subadditive. The main open problem here is whether static non-uniform strategies can give polylogarithmic approximation even when the number of buyers is large and they have subadditive valuations. A broader question is to ﬁnd better bounds on the maximum revenue any pricing strategy can obtain. We have used optimal social welfare as our bound, but even with one buyer, there are instances where the maximum social welfare and the maximum revenue that can be obtained by any strategy, is separated by a logarithmic factor. One may expect that if we instead try to design eﬃciently computable strategies and compare them against an optimal pricing strategy (or an optimal strategy in some broad class of strategies), better approximation factors, such as a constant, may be achieved.

112 Chapter 5

Sequential Posted Pricing in Multi-Unit Auctions

In this chapter, we study multi-unit auctions in Bayesian setting. We consider the following Sequential Posted Pricing problem in a K-unit auction. There is a single seller with K identical copies of a single item to sell, to n prospective buyers. Each buyer is interested in exactly one copy of the item, and has a value for it that is unknown to the seller. The buyers arrive in a sequence, and each buyer appears exactly once. The arrival order may be chosen by the seller. The seller quotes a price for the item to each arriving buyer, and may quote different prices to different buyers. Assuming that buyers are rational, a buyer buys the item if the price quoted to him is less than his value for the item, and pays the quoted price to the seller. This process stops when either K buyers have bought the item or when all buyers have arrived and left. Results in this chapter were published in collaboration with Eyal Even-Dar, Sudipto Guha, Yishay Mansour and S Muthukrishnan [20]. We focus on pricing and ordering strategies in the above model, called sequential posted- price mechanisms (SPMs), that maximize the seller’s expected revenue. A (non-adaptive) SPM is also equivalent to asking prices in parallel, and choosing to sell the items to up to K of the buyers who accepted their respective offered prices. Posted price mechanisms are clearly incentive compatible, and commonly used in practice. We design strategies in a Bayesian framework, where each buyer draws his value of the item from a distribution. These value distributions are known to the seller, and are used in designing the mechanism.

113 SPMs were recently studied in the general context of Bayesian single-parameter mechanism design (BSMD), which includes our K-unit auction, by Chawla et. al. [28]. They designed eﬃciently computable SPMs for various classes of BSMD problems and compared their expected revenue to that of the optimal auction mechanism, which was given by Myer- son [76]. For the K-unit auction, they showed that their SPM guarantees (1 1/e)-fraction − of the revenue obtained by Myerson’s auction. Bhattacharya et. al. [13] (as well as [28]) also used sequential item pricing to approximate optimal revenue, when the seller has multiple distinct items. However, the SPM computed by their algorithms may not be the optimal SPM, i.e. there may exist SPMs with greater expected revenue. Given that sequential pricing is quite common in practice, we focus in this chapter on eﬃciently computing an optimal SPM.

Our Results The results in [28] immediately imply a (1 1/e)-approximation for the − problem of computing optimal SPMs in K-unit auction. We strictly improve this bound. We design two diﬀerent algorithms – the ﬁrst is a polynomial time algorithm that yields a

K (1 K ) (1 1 )-approximation, and is meant for large values of K, and the second − K!eK ≈ − √2πK is a polynomial time approximation scheme (PTAS) for constant K. Combining these two algorithms yield a polynomial time approximation scheme for the optimal SPM problem, 1 for all values of K:ifK>2π2 , run the ﬁrst algorithm, else run the second algorithm. Recall that a PTAS is an algorithm that, for any given constant > 0, yields (1 )- − approximation in polynomial time (the exponent of the polynomial should be a function of only, and independent of input size). Note that a sequential posted pricing strategy can be adaptive – it can alter its prices and the ordering of the remaining buyers based on whether the current buyer buys the item. We shall call such strategies as Adaptive SPMs, or ASPMs, while SPM shall refer to a non-adaptive pricing and ordering strategy. Clearly, the expected revenue from an optimal ASPM is at least that from an optimal SPM. Further, the analysis of our ﬁrst algorithm shows that it gives the said approximation

K guarantee of (1 K ) (1 1 ) even against the expected revenue of Myerson’s − K!eK ≈ − √2πK revenue-optimal auction. Thus the gap between an optimal SPM and an optimal auction vanishes solely as a function of the inventory size is increased, and this result holds even if

114 the buyers’ pool is chosen adversarially based on K. In other words, a seller who owns a large inventory may commit to use a pricing strategy instead of an auction, and be assured that this commitment can lead only to a small regret, even before gathering any knowledge about the buyers’ pool. We design a third algorithm that outputs an ASPM, and is a PTAS for computing an optimal ASPM, for constant K. Again, combining this result with our ﬁrst algorithm, we obtain a PTAS for the optimal ASPM problem, for all values of K. Adaptive PTAS with multiplicative approximation is rare to ﬁnd in stochastic optimization problems. For example, an adaptive PTAS for the stochastic knapsack problem has been developed very recently [12]. The theorem below summarizes our results.

Our Techniques The first algorithm is based on a linear programming (LP) relaxation of the problem, such that the optimal solution to the LP upper bounds the expected revenue from any truthful mechanism, and in fact any Bayes-Nash equilibrium of any, possibly non- truthful, mechanism. We show that this LP has an optimal integral solution (after adding infinitesimal perturbations to break ties arbitrarily), from which we construct a pricing for the buyers. The buyers are ordered simply in decreasing order of prices – it is easy to see that this is an optimal ordering policy given the prices. The LP formulation implies that if there were no limit on the number of copies the seller can sell, then the expected revenue obtained from this pricing would be equal to the LP optimum, and at most K copies of the item are sold in expectation. However, the algorithm is restricted to selling at most K copies in all realizations, and the result follows by bounding the loss due to this hard constraint. The interesting property we find is that this loss vanishes as K increases. It should be noted that an LP-based approach is used in [13]; however, they consider a more general problem with multiple distinct items, and their analysis yielded no better than constant approximation factors. The second algorithm uses a dynamic programming approach, which is common in the design of approximation schemes. We make some key observations that reduce the problem to an extended version of the generalized assignment problem (GAP) [82, 29] with constant number of bins, which has polynomial time algorithm (polynomial in the size of bins and number of items) using dynamic programming. The main observation is that in any SPM,

115 if we pick a contiguous subsequence of buyers to whom there is very small probability of selling even a single copy, and arbitrarily permute this subsequence, the resulting SPM will have almost the same expected revenue as the original SPM. This observation drastically cuts down the number of conﬁgurations that we have to check before ﬁnding a near-optimal SPM. The third algorithm for computing ASPM is a generalization of the second algorithm, but it must now approximate a decision tree, that may branch at every step based on whether a copy is bought, instead of an SPM sequence. The key observation in this case is that there exists a near-optimal decision tree that does not branch too often, and the problem again reduces to an extension of GAP with constant number of bins.

Other Related Work Maximizing welfare via truthful mechanisms in prior-free settings have been studied for K-unit auctions [44, 45] and combinatorial auctions, such as the work described in Chapter 4. Bayesian assumptions has led to constant approximation against optimal revenue from any auction for special cases of combinatorial auctions [13, 28]. But Bayesian assumptions can also lead to tighter upper bounds on optimal sequential pricing, and that is one of our main contribution. A parallel posted-price approach has been used in a more complex repeated ad auction setting to get constant approximation [21].

5.1 Preliminaries

In a K-unit auction, there is a single seller who has K identical copies of a single item, and

wish to sell these copies to n prospective buyers B1,B2 ...Bn. Each buyer Bi is interested

in one copy of the item, and has value vi for it. vi is drawn from a distribution speciﬁed

by cumulative distribution function (cdf) Fi that is known to the seller. The values of diﬀerent buyers are independently drawn from their respective distributions. Without loss of generality, we assume that K n. ≤

Deﬁnition 5.1.1. Let piv denote the probability that Bi has value v for the item. Let p˜iv

denote the probability that Bi has value at least v.Weshallcallitthesuccess probability

when Bi is oﬀered price v.Clearlyp˜iv = v v piv . ≥

116 We assume, for all our results, that each value distribution is discrete, with at most L distinct values in its support (i.e. these values have non-zero probability mass). Let UVi be n the support set of values for the distribution of Bi, and let UV = i=1 UVi . We shall also 1 assume that L is polynomial in n, and thatp ˜iv is an integral multiple of 10n2 for all i, v. These assumptions are without loss of generality for obtaining PTAS for optimal SPM or ASPM (see Section 5.4.1 for a brief discussion).

Definition 5.1.2. A sequential posted-price mechanism (SPM) is a mechanism which considers buyers arrive in a sequence, and offers each of them a take-it-or-leave-it price: the buyer may either buy a copy at the quoted price or leave, upon which the seller makes an offer to another buyer. Each buyer is given an offer at most once, and the process ends when either all K copies have been sold, or there is no buyer remaining. An SPM specifies the entire sequence of buyers and prices before the process begins. In contrast, an adaptive sequential posted-price mechanism (ASPM) may decide the next buyer based on which of the current and past buyers accepted their offered prices.

Note that there can be no adaptive behavior when K = 1, since the process stops with the first accepted price. Thus an ASPM can be specified by a decision tree: each node of the tree contains a buyer and a price to offer. Each node may have multiple children. The selling process starts at the root of the tree (i.e. offers the price at the root to the buyer at the root), and based upon whether a sale occurs at the root, moves to one of the children of the root, and continues inductively. The process stops when either K items have been sold, or n buyers have appeared on the path in the decision tree traversed by the process – the latter nodes are the leaves of the decision tree. It is easy to see that the decision of an optimal ASPM at any node of the tree should depend only on the number of copies of the item left and the remaining set of buyers (the latter is solely determined by the node reached by the process). Thus, each node has at most K children, at most one each for the number of copies left. Note that an ASPM may not adapt immediately to a sale – it may move to a fixed buyer regardless of the outcome. Such a node will only have a single child. Without loss of generality, we shall represent an ASPM such that each non-leaf node either has a single child or K children (some of which may even be infeasible). The latter nodes are called branching nodes. In this context, an

117 SPM is simply an ASPM whose decision tree is a path.

Note that SPM and ASPM are incentive compatible: a buyer Bi buys the item if and only if its value vi is equal to or greater than the price oﬀered to it, and pays only the quoted price to the seller.

Deﬁnition 5.1.3. The revenue R(v1,v2 ...vn) obtained by the seller for a given SPM is the sum of the payments made by all the buyers, which is a function of the valuations of the buyers. The expected revenue of an SPM or ASPM is computed over the value distributions

Ev F [R(v1, v2 ...vn)].Anoptimal SPM or ASPM is an SPM (respectively, ASPM) that i∼ i gives the highest expected revenue among all SPMs (respectively, ASPMs). Let the expected revenue of an optimal SPM (or ASPM) be OPT.Anα-approximate SPM (or ASPM, respectively), where α 1, has expected revenue at least αOPT. ≤

5.1.1 Basic Result

An SPM must specify an ordering of the buyers as well as the prices to oﬀer to them. It is worth noting that if either one of these tasks is ﬁxed, the other task becomes easy.

Lemma 5.1.1. Given take-it-or-leave-it prices to oﬀer to the buyers, a revenue-maximizing SPM with these prices simply considers buyers in the order of decreasing prices. Given an ordering of buyers, one can compute in polynomial time a revenue-maximizing ASPM that uses this ordering (and only adapts the oﬀered prices).

Proof. For the first claim, consider an SPM where there are two buyers Bi and Bj,such that Bi arrives just before Bj,butisoffered a lower price than Bj. Consider the modified

SPM created by swapping Bi and Bj in the order, while keeping the price offered to Bi,Bj and other buyers unchanged. In realizations where at most one of Bi or Bj accepts the price, the revenue of the original and modified SPMs are equal. However, in realizations where both buyers accept their offered prices, the selling process may not reach the latter buyer, and so the modified SPM has higher or equal revenue in that case. For the second claim, we can compute the prices using dynamic programming. Let the buyers be ordered as Bπ(1),Bπ(2) ...Bπ(n). Let A(i, j) denote the maximum expected revenue that can be obtained from the last i buyers in the given ordering, if there are j items

118 left to sell to them. For initialization, set A(1, 0) = 0, and A(1,j) = maxx U xPr [vn x] ∈ V ≥ for j 1, which is the maximum expected revenue from B with the item in stock. Suppose ≥ n A(i 1,j) has been computed for all j.ThenA(i, j) can be computed by iterating through − all possible prices to oﬀer Bn i+1, and pick one that yields highest expected revenue. For − a price x UV ,theexpectedrevenueis(x + A(i 1,j 1))Pr [vn i+1 x]+A(i ∈ − − − ≥ − 1,j)Pr [vn i+1

5.2 LP-based Algorithm for Large K

In this section we present our ﬁrst algorithm that yields us an approximation factor that improves as K increases, and implies a vanishing adaptivity gap. The following theorem summarizes our result.

Theorem 14. For all K 1, if a seller has K units to sell, there exists an SPM whose ≥ K expected revenue is at least 1 K 1 1 fraction of the expected revenue of an − K!eK ≥ − √2πK optimal auction. This SPM can be computed in polynomial time.

As a first step to our algorithm, we add random infinitesimal perturbation to the values v U and the associated probability values p , so that almost surely, U are disjoint, ∈ Vi iv Vi and further, all the values and probabilities are in general position. Intuitively, this property is used in our algorithm to break ties. By Myerson’s characterization [76], any truthful mechanism (in particular, Myerson’s optimal auction) can be viewed as follows: it sets a price for each buyer Bi as a function of the bids of all other buyers, and Bi receives the item if and only if its valuation exceeds this offered price. Consider the event Eiv that Bi is offered the item at price v, and accepts the offer. Let y denote the probability of that E occurs when is implemented. Let iv iv P x denote the probability that B was offered price v when is implemented. Note that iv i P both probabilities are taken over the value distributions of the buyers, as well as internal randomization of . Naturally, we must have y p˜ x . Also, by linearity of expectation, P iv ≤ iv iv n n the expected revenue obtained by is i=1 v U vyiv. Moreover, i=1 v U yiv is the P ∈ V ∈ V expected number of copies of the item sold by the seller, and this quantity must be at most

119 K. Finally, the mechanism enforces that each buyer is oﬀered a price at most once in any realization, and hence in expectation,i.e. v U xiv 1. ∈ V ≤ Viewing xiv and yiv as variables depending upon the mechanism used, optimum of the following linear program Lp-K-SPM provides an upper bound to the expected revenue from any auction. Our algorithm involves computing an optimal solution to this program with a speciﬁc structure, and use the solution to construct an SPM. We also consider its dual program, Dual-K-SPM.

Lp-K-SPM = max vyiv i=1 v UV Dual-K-SPM =minKτ + λ ∈ i i yiv p˜ivxiv i [1,n],v UV ≤ ∀ ∈ ∈ ζ + τ v iv ≥ x 1 i [1,n] v UV iv ∈ ≤ ∀ ∈ λi v p˜ivζiv 0 n − ≥ i=1 v U yiv K ∈ V ≤ ζ , λ , τ 0 iv i ≥ y ,x 0 iv iv ≥

Lemma 5.2.1. Assuming that the points in UVi and the probabilities p˜iv have been perturbed inﬁnitesimally, and so are in general position, there exists an optimal structured solution xiv∗ ,yiv∗ of Lp-K-SPM, computable in polynomial time, such that:

1. for all i, v, yiv =˜pivxiv.

2. for each i there is exactly one v such that xiv > 0. Let v(i) denote the value for which

xiv(i) > 0.

3. There exists at most one i such that 1 i n and x =1. If such i = i exists, ≤ ≤ iv(i) n then v(i)=mini=1 v(i).

Proof. Given any feasible solution to Lp-K-SPM,whereyiv < p˜ivxiv for some i, v, we can simply reduce xiv till yiv becomes equal top ˜ivxiv. This change keeps the solution feasible, and also leaves the objective unchanged. So we can simply eliminate the variables yiv from

Lp-K-SPM by setting yiv =˜pivxiv. An optimal solution to this modified LP will also be an optimal solution for the original LP, and naturally satisfy the first condition in the lemma. By a minor abuse of notation, we refer to this modified LP as Lp-K-SPM.

120 Let us now consider the Lagrangian program LagrangianSPM(τ) obtained by removing the constraint of selling at most K copies, and associating a cost τ of violating this

constraint in the objective. The following property holds by LP duality: let τ ∗ be the assignment to variable τ in an optimal solution to Dual-K-SPM. Then the optimum of

LagrangianSPM(τ ∗) is equal to the optimum of Lp-K-SPM in value. We shall compute an

optimal solution of LagrangianSPM(τ ∗) that is also feasible for Lp-K-SPM, and satisﬁes

either τ ∗ = 0 or i v p˜ivxiv = K. Such a solution must also be an optimal solution of Lp-K-SPM.

LagrangianSPM(τ) = max vp˜ x + τ(K p˜ x ) = Kτ + max (v τ)˜p x iv iv − iv iv − iv iv i v i v i v x 1 iv ≤ v y ,x 0 iv iv ≥

Ifp ˜iv = 0, then we assume that xiv is set to zero, since this does not aﬀect feasibility or

value of the objective. Let an optimal solution of LagrangianSPM(τ ∗) be denoted by xiv∗ (τ ∗). Such a solution must satisfy x (τ ) = 0 for all v<τ . Further for some i, if max (v τ )˜p iv∗ ∗ ∗ v − ∗ iv

is maximized at a unique v,thenxiv∗ (τ ∗)=1ifv = arg maxv UV (v τ ∗)˜piv v τ ∗ , and 0 ∈ i { − | ≥ } otherwise. If max (v τ )˜p > 0, then the added perturbations ensure that the maximum v − ∗ iv is indeed unique. Suppose the maximum is zero for some i,then˜p = 0 and so x (τ )=0 v>τ .Since iv iv∗ ∗ ∀ ∗ every buyer has some non-zero probability of having a positive value for the item (else we

can simply neglect such buyers), we have τ ∗ > 0. The only xiv∗ (τ ∗) that we may set to a non- zero value is for v = τ , provided that τ U . Because of added perturbations, this can ∗ ∗ ∈ Vi happen for at most one buyer i. We ﬁrst ﬁx the assignment of all other variables as described above, then set this x to the highest value less than 1 such that p˜ x ) K.This iv∗ i v iv iv ≤ gives our required structured solution. Given τ ∗, constructing the solution requires linear

time. As mentioned before, τ ∗ can be computed by solving Dual-K-SPM; one may also use binary search techniques for this purpose, similar to many packing LPs (details omitted, see, eg. [80]).

121 Our algorithm for computing an SPM is as follows:

1. Compute an optimal structured solution of Lp-K-SPM.

2. In the SPM, oﬀer price v(i)toBi, and consider buyers in order of decreasing v(i).

5.2.1 Approximation Factor

It remains to analyze the approximation factor of our algorithm. Let the order of decreasing prices be B ,B ...B . For 1 i

xπ∗(n)v(π(n)) in the structured optimal solution may not have been 1, so let Zn be v(π(n)) = zn n with probability xπ∗(n)v(π(n))p˜π(n)v(π(n)) = un and 0 otherwise. If Z = i=1 Zi,thenE[Z] is the optimum of the LP solution. The revenue of the algorithm, however, is at least equal to

the sum of the ﬁrst K variables in the sequence Z1,Z2 ...Zn that are non-zero. Let this sum be denoted by the random variable Z . Note that z z ... z , and n u K. 1 ≥ 2 ≥ ≥ n i=1 i ≤ The following lemma immediately implies Theorem 14.

K Lemma 5.2.2. E[Z ] (1 K )E[Z] (1 1 )E[Z]. ≥ − K!eK ≥ − √2πK

Proof. Let α(i)=ziui. Let the probability that we reach Zi in the sequence before ﬁnd- ing K non-zero variables, be given by the function f(i, u) (this function is independent n of z1,z2 ...zn), where u =(u1,u2 ...un). Then E[Z]= i=1 f(i, ˜u ) α(i), while E[Z] = n =1 α(i). Observe that f(i, u) is monotonically decreasing ini. We shall narrow down the

the instances on which E[Z]/E[Z] is minimized.

Claim 3. Given an instance comprising variables Z1,Z2 ...Zn such that zi >zi+1,onecan

modify it to construct another instance Z˜1, Z˜2 ...Z˜n such that E[Z]/E[Z] decreases.

Proof. We modify the instance by deﬁning α (j) for 1 j n, and setting the possible ≤ ≤ non-zero value of Z˜j to bez ˜j = α(j)/uj, with success probability remaining uj:

α(j)ifj = i, i +1 zi zi+1 α(j)= α(i) ∆ if j = i where ∆ = − > 0  1 + 1  − u(i) u(i+1)  α(i + 1) + ∆ if j = i +1  

122 Note that Z˜ = Z j = i, i + 1, so only Z and Z gets modiﬁed. Further,z ˜ are non- j j ∀ i i+1 j increasing in j (in fact,z ˜i =˜zi+1) so the modiﬁed instance is valid. Also, i α(i)= i α(i), so E[Z] remains unchanged. u remains unchanged too, and hence the probabilities f(i, u). Finally, the change in E[Z ]is(f(i +1, u) f(i, u)) ∆ < 0, i.e. E[Z ] decreases. −

Thus, we can restrict our attention to instances where z1 = z2 = ... = zn = z∗ (say).

Without loss of generality, we let z = 1, so that Z1,Z2 ... are Bernoulli variables, and Z =min Z, K . Note that the ordering of the variables do not inﬂuence Z .Thenextstep { } is to show that if we split the variables, keeping E[Z] unchanged, E[Z] can only decrease.

Claim 4. Let Z1,Z2 ...Zn be Bernoulli variables, such that the success probability is

Pr [Zj = 1] = uj. Suppose that we modify the set of variables by removing Zi from it and adding two Bernoulli variables Z˜i and Zˆi to it, where Pr Z˜i =1 =˜ui > 0 and

Pr Zî =1 =ûi > 0,andu˜i +ûi = ui. Then E[Z]=E[min Z,K] decreases or remains unchanged due to this modification, while E[Z] = K remains unchanged.

i 1 n Proof. Let X be the sum of the remaining variables, i.e. X = j−=1 Zj + j=i+1 Zj.We shall show that E[Z X K 2] and E[Z X K] remain unchanged by the modification, | ≤ − | ≥ while E[Z X=K 1] decreases, thus proving . | − If X K 2, then Z is X +Z in the original instance, and X +Z˜ +Zˆ in the modified ≤ − i i i instance. Since E[Z ]=E[Z˜ + Zˆ ]=u,soE[Z X K 2] remains unchanged. Also, if i i i i | ≤ − X K,thenZ is simply K in both instances. If X = K 1, then Z = K 1+Z ≥ − − i in the original instance and Z = K 1 + min 1, Z˜ + Zˆ in the modified instance. So − { i i} E[Z X=K 1] = K 1+u and E[Z K 1] = K 1+Pr Z˜ + Zˆ 1 =K 1+(˜u + | − − i | − − i i ≥ − i û ˜u û ) < K 1+u,respectively. i − i i − i Assume that the success probabilities of the Bernoulli variables are all rational – since rational numbers form a dense set in reals, this shall not change the lower bound we are seeking. Then, there exists some large integer N such that all the probabilities are integral multiples of 1/N . Further, we can choose an arbitrarily large N for this purpose. Now, split each variable that has success probability t/N into t variables, each with success probability

1/N . The above claim implies that E[Z]/E[Z] can only decrease due to the splitting. Thus,

123 it remains to lower bound E[Z ]/K for the following instance, as N : KN Bernoulli →∞ variables, each with success probability 1/N . For this final step, we use the well-known property that the sum of Bernoulli variables with infinitesimal success probabilities approach the Poisson distribution with the same mean. In particular, if P is a Poisson variable with mean K, then the total variation distance between Z and P is at most (1 e K )/N (see e.g. [10]), which tends to zero − − as N .Thus,wesimplyneedtofindE[min P, K]/K, and this is the lower bound on →∞ K E[Z ]/E[Z] that we are seeking. It can be verified that E[min P, K] = K(1 K )(see − K!eK Section 5.4.2), which proves the lemma.

5.3 PTAS for constant K

We now deﬁne an optimization problem called VersionGAP, and our PTAS for both SPM and ASPM for constant K will reduce to solving multiple instances of this problem.

VersionGAP: Suppose there are n objects, and each object has L versions. Let version j of object i have proﬁt p and size s 1. Also, suppose there are C bins 1, 2 ...C,where ij ij ≤ bin has size s and a discount factor γ . The goal is to place versions of objects to bins, such that:

1. Each object can be placed into a particular bin at most once, as a unique version. If

object i is placed as version j into bin , then it realizes a proﬁt of γpij and a size of

sij.

2. Each object can appear in multiple bins, as diﬀerent versions. However, there is a

given collection FC of feasible subsets of bins 1, 2 ...C. The set of bins that an object is placed into must be a feasible subset.

3. The sum of realized sizes of objects placed into any bin must be less than s.

The profit made by an assignment of object version to bins, that satisfy all the above conditions, is the sum of realized profits by all objects placed in the bins. The goal is to find an assignment that maximizes the profit.

124 Lemma 5.3.1. For all objects and versions i, j, let sij be a multiple of 1/M for some ﬁxed M 2. Then an optimal solution to VersionGAP can be found in time (ML)O(C)n. ≥ Proof. The algorithm is a simple dynamic programming. Order the objects arbitrarily. Let

D(i, j1,j2 ...jC ) be an optimal feasible assignment (or the profit thereof, by an abuse of notation) of the first i objects, such that the sum of realized sizes of objects in bin is j , for =1, 2 ...C. D(i, j ,j ...j ) is assigned as null and its profit as if no such 1 2 C −∞ assignment exists. Note that we only consider j to be multiples of 1/M and at most 1, for all .

D(0,j1,j2 ...jC ) is null for all j1,j2 ...jC , except for D(0, 0, 0 ...0) which is zero. Suppose D(i 1,j ,j ...j ) have been computed for all j ,j ...j . Then to compute − 1 2 C 1 2 C D(i, j ...j ), we ﬁrst choose a feasible subset of bins from to place it in ( < 2C 1 C FC |FC | choices), then its version in each bin in this subset (at most LC choices), and then compute the objective as D(i 1,j s ,j s ...j s )+ C γ p ,wheret is the version − 1 − it1 2 − it2 C − itC =1 it in which object i is chosen to be placed in bin (if the object is not placed in bin , treat

sit and pit as zero). We iterate through all the choices to maximizes this objective. Thus, computing each C C entry D(i, j1 ...jC ) takes time at most O(C(2L) ). The number of entries is at most nM . The maximum among all the entries gives the required assignment.

5.3.1 PTAS for Computing SPM

We now design an algorithm to compute a near-optimal SPM for constant K.

Theorem 15. There exists a PTAS for computing an optimal SPM, for any constant K. poly(k, 1) The running time of the algorithm is nk − , and gives (1 )-approximation. − We shall, without loss of generality, give a (1 ck)-approximation, and this will imply − the above theorem: putting = /ck will yield a (1 )-approximation. − We first establish some definitions that we shall use. Let a segment refer to a sequence of some buyers and prices offered to these buyers – we shall refer to parts of an SPM as segments. Let the undiscounted contribution (B ) of a buyer B ,whenoffered price x(B ), V i i i

be α(Bi)=x(Bi)˜pix(Bi),whileitsweight bep ˜ix(Bi), its success probability. Undiscounted

125 contribution (S) of a segment S is the sum of undiscounted contributions of buyers in the V segment, and the weight of the segment is the sum of their weights. Given an SPM, let dis(B) denote the probability that the selling process reaches buyer B.Thereal contribution of a buyer to the expected revenue is α(B)dis(B), and the expected revenue of the SPM is the sum of the real contributions of all the buyers. More generally, let γ(B) denote the probability that Bi is reached with at least items remaining. Then dis(B)=γ1(B). The discount factor dis(S) of a segment S, whose first buyer is B,is defined to be dis(B). Similarly, we define γ(S)=γ(B). We present our algorithm through a series of structural lemmas, each of which follows quite easily from the preceding lemmas. The first step towards our algorithm is that we can restrict our attention to truncated SPMs.

K Lemma 5.3.2. There exists an SPM of total weight at most K log , where each buyer has discount factor at least , that gives an expected revenue of at least (1 )OPT.Weshall − refer to SPMs that satisfy this condition as truncated.

Proof. Consider the smallest prefix of the optimal SPM (with expected revenue OPT)such that the discount factor of the corresponding suffix, obtained by removing the prefix, is at most . Moreover, if we were to simply omit this prefix, then the expected revenue of the remaining segment can at most be OPT. So the contribution of the remaining segment to the optimal SPM is at most OPT, and the prefix alone has expected revenue expected revenue at least (1 )OPT. By Fact 3, the probability that no copy gets sold in a segment − K K of weight log is at most /K. Thus the weight of the prefix is at most K log .

We can now restrict ourself to approximating an optimal truncated SPM. The following deﬁnition of a permutable segment will be crucial to the description of our algorithm.

Deﬁnition 5.3.1. We shall call an SPM segment permutable if either:

3 1. its weight is at most δ = 20K3 . We shall refer to such a permutation segment as a small buyers segment.

2. it has a single buyer, possibly of weight more than δ. In this case, we shall refer to this buyer as a big buyer.

126 Any SPM can clearly be decomposed into a sequence of permutable segments and big buyers. Moreover, any truncated SPM can be decomposed into a sequence of at most K K log C = O( δ ) permutable segments. This is because if the permutable segments are maximally chosen, then two consecutive permutable segments in the decomposition either have at least one big buyer between them, or their weights must add up to more than δ (otherwise, the two segments can be joined to create one permutable segment).

Fact 3. Let 1 >y ,y ...y > 0. Let y = s. Then 1 s + s2 >e s > (1 y ) > 1 2 j=1 j − − j=1 − j 1 s. −

Lemma 5.3.3. The probability of selling at least one copy of the item in a small buyers permutable segment that has weight s is at least s s2. The probability of selling at least − t 1 copies (assuming that at least t copies are left as inventory) in such a segment is at ≥ most st. So the probability of selling exactly one copy is at least s 2s2. − Proof. Fact 3 implies that the probability of selling at least one item is at least s s2 and − at most s. For the second statement, consider t = 2. Conditioning on a particular buyer B in the segment buying a copy, the probability that the remaining buyers in the segment buy at least one copy is at most s. The two events are independent, so the probability of their simultaneous occurrence is the product of their probabilities. Summing over all buyers in the segment, we get that the probability that at least two items are bought is at most s2. The argument scales in a similar fashion for higher values of t: probability that t items are bought is at most st.

Lemma 5.3.4. Consider a permutable segment of weight s appearing in an SPM, and let its discount factor be γ. Then the discount factor of the last buyer in the segment is at least γ(1 s). If the undiscounted contribution of the segment is α, then the real contribution of − buyers in this segment to the expected revenue is at least αγ(1 δ) and at most αγ. − Proof. The probability of the process not stopping inside the segment, conditioned upon reaching it, is at least the probability of not selling any copy in the segment, which is at least 1 s (it is exactly 1 s for a big buyer segment). − − 127 The above lemma shows that the real contribution of a segment can be approximated by the product of its discount factor and its undiscounted contribution, which does not depend on the exact buyers, their relative ordering or prices in that segment. We next show that the discount factor of a segment, given a decomposition of an SPM into permutable segments, can also be approximated as a function of the approximate sizes of preceding segments.

Lemma 5.3.5. Given an SPM, that can be decomposed into an ordering of permutable

segments S1,S2 .... Let Si be a small buyers segment. Let s be the weight of Si. Then γ (S )(1 s)+γ (S )s +4s2 γ (S ) γ (S )(1 s)+γ (S )s 2s2. i − +1 i ≥ i+1 ≥ i − +1 i − Proof. Directly using the bounds in Lemma 5.3.3 to the formula: γ (S )= K γ (S )Pr [Exactly (j )copies of the item are bought by buyers in S ]. i+1 j= j i − i

The lemma below follows easily from Lemma 5.3.5.

Lemma 5.3.6. Given any SPM decomposed into Q C permutable segments S ,S ..., ≤ 1 2 such that the weight of S is between s + τ and s τ for all 1 i n , where τ = δ/20C. i i i − ≤ ≤ th Consider an alternate SPM (with possibly diﬀerent buyers), that has n buyers, and the i th buyer in the segment has weight si. Let ρ(,i) be the probability that the i buyer is reached in the alternate SPM with at least items remaining. Then

ρ(,i) 12(δ2 + τ)i) γ (S ) ρ(,i) + 12(δ2 + τ)i) . − ≤ i ≤

3 If the SPM is truncated, then dis(S )=γ (S ) ,andsincei Q C, δ = and i 1 i ≥ ≤ ≤ 20K3 τ δ/20C, so we can get a multiplicative guarantee ρ(1,i)(1 ) dis(S ) ρ(1,i)(1+) . ≤ − ≤ i ≤ We shall refer to the following as a configuration: An ordering of up to C permutable segments, where each permutable segment is specified only by the weight of the segment and δ big buyer respectively, each weight being a multiple of τ = 20C . Note that the configuration does NOT specify which buyer belongs to which segment, or the individual weights of the buyers. This is because a configuration is specified by at most C positive integers (weight 1 of each segment is specified by a positive integer z<τ , which indicates that the weight is

zτ). We shall represent a conﬁguration z as an ordered tuple of integers (z1,z2,z3 ...). Note

128 1 O(C) K O(K) that there are at most ( τ ) =( ) distinct configurations. We say that an SPM has configuration z if it can be decomposed into an ordering of permutable segments S1,S2 ... such that S has weight at least (z 1)τ and at most z τ. i i − i For any given configuration z, the expected revenue of an SPM with configuration z can be approximated, up to a factor of (1 δ)(1 2) by a linear combination of the undiscounted − − contribution of the permutable segments, where the coefficients of the linear combination depend only on z.Thecoefficients are the discount factors, which can be computed by looking at an alternate SPM with a buyer for each segment, such that the ith buyer has weight ziτ. This is a direct conclusion of Lemma 5.3.6 and Lemma 5.3.4. The discount factors of each buyer in the alternate SPM can be easily computed in O(CK)timeusing th dynamic programming. Let Az(i) denote the discount factor of the i buyer in the alternate SPM corresponding to z. For any configuration z, we compute prices for the buyers, and a division of buyers into permutable segments S ,S ...such that S has weight at most z τ, and A (i) (S ) 1 2 i i i z V i is maximized (it is not necessary to include all buyers). This is precisely an instance of VersionGAP, where each buyer is an object, the different possible prices and the corresponding success probabilities create the different versions, and the sizes of the bins are given by z, and the feasible subsets for an object simply being that each object can get into at most one bin. This can be solved as per Lemma 5.3.1. The solution may not saturate every bin, and hence may not actually belong to configuration z. However, for any two configurations z =(z ,z ,z ) and z =(z ,z ...z ), such that z z 1 i t,wehave 1 2 t 1 2 t i ≤ i ∀ ≤ ≤

Az(i) >Az (i). So the SPM formed by concatenating S1,S2 ... in that order generates revenue at least (1 3) times the revenue of the optimal sequence that has configuration − z. Thus our algorithm is to find an SPM for each configuration, using the algorithm for VersionGAP, and output the best SPM among them as the solution.

5.3.2 PTAS for Computing ASPM

We now design an algorithm to compute a near-optimal SPM for constant K.

Theorem 16. There exists a PTAS for computing an optimal SPM, for any constant K.

129 (k 1)O(k) The running time of the algorithm is nk − , and gives (1 )-approximation. − As mention in Section 5.1, an ASPM is specified by a decision tree, with each node containing a buyer and an offer price. We extend some definitions used for SPMs to ASPMs. The weight of a node is the success probability at this node conditioned on being reached. A segment in an ASPM is a contiguous part of a path (that the selling process might take) in the decision tree. A segment is called non-branching if all but possibly the last node are non-branching. Other definitions such as weight and contribution of a segment are identical. A permutation segment is a non-branching segment satisfying properties as defined earlier (Definition 5.3.1). The discount factor of a node (or a segment starting at this node, or a subtree rooted at this node) is the probability that the node is reached in the selling process. Consider any ASPM whose tree is decomposable into D non-branching segments, each of weight at most H. (Note that D = 1 for an SPM.) Then the entire tree of a truncated ASPM decomposes into C = O(DH/δ) permutable segments. We shall refer to such ASPMs as C-truncated ASPMs. A configuration for a C-truncated ASPM shall now list the weights of at most C permutable segments and also specify a tree structure among them, i.e. the parent segment of each segment in the decision tree. Moreover, since each path can have no more than C segments, it is sufficient to specify the weights to the nearest multiple of τ = δ/20C, to get the discount factor of each segment with sufficient accuracy. So there are (C/τ)O(C) = CO(C) configurations for C-truncated ASPMs. For each configuration, we can use VersionGAP to compute an ASPM that is at least (1 ) times the revenue of an optimal ASPM with that configuration, as before. Each − VersionGAP instance has C bins in this case. The discount factor of each permutable segment in the configuration can be computed with sufficient accuracy, similar to Lemma 5.3.6. Iterating over all possible configurations, we can find a near-optimal C-truncated ASPM. Solving VersionGAP requires time exponential in the number of bins (see Lemma nkC O(C) 5.3.1), so the entire running time of the above algorithm is ). The problem is that for the above algorithm to be a PTAS,C must be a function of K and 1 − only. Lemma 5.3.7 achieves this goal through a non-trivial structural characterization, and immediately implies Theorem 16.

Lemma 5.3.7. There exists an ASPM with the following properties:

130 1. Its expected revenue is at least (1 ) times the expected revenue of the optimal ASPM. − 2. The decision tree is decomposable into D =(K/)O(K) non-branching segments.

3. Each non-branching segment in the tree has weight at most H =(K/)O(1).

4. Each path in the tree consists of at most (K/)O(1) permutable segments.

Proof. Let us view an optimal ASPM decision tree, with expected revenue OPT as consisting of a spine, which is the path followed if no buyer buys a copy, along with decision subtrees hanging from many, possibly all, nodes of the spine. Note that all nodes may not be branching nodes, so a spine need not be left by the process at the very moment that a sale is recorded, but may branch out at a later point. Each such subtree, hanging from a node w (say) on the spine, are optimal ASPMs for selling some

there are at most (K/)O(1) branching nodes on the spine, and • the weight of the spine shall be at most (K/)O(1), • while only losing a factor of (1 c) in expected revenue for some constant c. − The subtrees, since they are selling less than K copies, can be transformed inductively (when a single copy is left, the subtree is just a path and trivially satisfies the required properties). Such a tree will satisfy the properties listed in Lemma 5.3.7 (for the last property, note that any path can be decomposed into at most K contiguous parts, each of which is a spines of some subtree, since leaving a spine implies a sale). Overall, the entire transformation shall cause a loss factor of (1 cK). This achieves our goal, since we could − have instead started by scaling down to /cK. As a first step, we truncate the spine. For any node w,letR(w)betheexpectedrevenue obtained from the rest of the selling process (excluding the contribution of the buyer at w itself), conditioned upon the selling process reaching node w. We find the earliest (i.e. closest to the root) node w on the spine such that R(w) OPT, and delete all children of ≤ w and the subtrees under them. This only causes a loss of OPT – moreover, the probability of reaching w could have been at most , so the weight of the truncated spine is at most

131 K K log (similar argument as Lemma 5.3.2). This immediately achieves the second property listed above, and it remains to limit the number of branching nodes. We can now assume that R(w) > OPT for all nodes w on the spine.

For a node w,letR(w) denote, conditioned upon the selling process reaching w and then have less than K items to sell after w, the expected revenue from the rest of the selling process. Clearly, R(w)

R(w) >R(w)/4K. This is because R(w) is the result of selling at least one copy of the item to the same set of buyers as R(w), except that R(w) may have as many as K copies of the item. Looking back at Section 5.2, if the number of items is decreased from K to 1 (keeping set of buyers unchanged), then the optimum of the linear program Lp-K-SPM decreases by a factor of at most K (scaling down the variables by a factor of K gives a feasible solution), and the optimal revenue is always within factor 1/2 of the LP optimum (since 1 1 1/2 for all K). This shows that R (w) > OPT/4K for all nodes on the − √2πK ≥ spine. Divide the spine into segments that either consist of a single buyer, or multiple buyers whose weights add up to no more than δ. These segments may have branching nodes in them, and hence may not be permutable. Clearly there are at most (K/)O(1) such segments, and now we shall focus on modifying each segment separately. We shall modify subtrees hanging from nodes in the segment, so that the segment can be subdivided into poly(K/) non-branching segments, thus completing the proof. Clearly we need to only consider those segments that comprise multiple small buyers. Let us consider one such segment, and describe the necessary modification to the tree. Define a minimal set of pivotal nodes in the segment, that satisfies the following condition: For any node w in the segment, there is a pivotal node v that is a descendant of w, such that R (v) (1 )R (w). Since OPT R (w) OPT for all nodes w, we have at ≥ − 4K ≤ ≤ 1 most O(− log(K/)) pivotal nodes. We shall make modifications to the decision tree so that the pivotal nodes are the only branching nodes in the segment. Let v be the pivotal node satisfying this condition for w, that is nearest to w in the segment. Suppose that w is a branching node. We delete all children of w that are not

132 part of the spine, and simply make it a non-branching node. We do this for all non-pivotal, branching nodes in the segment. Recall that at every branching node, the choice of which children the process follows is based only upon the number of copies of the item left. Now, the segment has few enough branching nodes – branching nodes are a subset of pivotal nodes.

To argue a limited loss in revenue, we need to analyze the values R(w)inthemodiﬁed trees, let us denote them by R (w). It suﬃces to show that R (w) (1 2)R (w). mod mod ≥ −

Since R(w1) and Rw 2 ,wherew1 and w2 are distinct nodes on the spine, are expectations conditioned upon disjoint events, this implies that the expected revenue of the entire tree falls by a factor of at most (1 2) due to this modification. − To show that R (w) (1 2)R (w), we can almost say that R (w) is at least to mod ≥ − mod R(v), since the branching has been deferred until node v. The only difference is that some small buyers get executed between w and v. So if there are items left after w,theremay be less than items when v is reached in the modified tree – however, the probability of this event is less than δ, and is independent of the history of events up to w. So, neglecting the contribution of nodes between w and v (but taking into account their discounting effect on descendant nodes), R (w) (1 δ)R (v). Since R (v) (1 )R (w), we have our mod ≥ − ≥ − result. 1 Thus each segment has at most O(− log(K/)) branching nodes now, which implies that the entire spine has (K/)O(1) branching nodes. This completes the proof.

5.4 Additional Details

5.4.1 Discretization

We explain why we can assume the following for the value distribution of each buyer: it is discrete, and the probability mass at all points, if non-zero, is an integer multiple of 1 . The assumption can only cause a loss of (1 1 ) in the expected revenue: given an n2 − n instance, we can create a discrete distribution with the above properties, corresponding to each value distribution, and an algorithm for computing an α-approximate SPM or ASPM in the modified instance gives an α(1 1 )-approximation for the original instance. − n Let civ = vp˜iv be the expected revenue from buyer Bi if price v is posted to it. First, we can simply keep only those v that are powers of (1 1 ), and assume that there is probability − n2 133 mass on only these points (leavep ˜iv unchanged). Next, for each such v, alter v andp ˜iv so that their product civ remains unchanged, butp ˜iv changes to the closest integral multiple of 1 n2 that is greater thanp ˜iv. This does not change the possible choices of expected revenue that can be obtained from a buyer upon reaching it, and their effect on future buyers, i.e. success probability, changes by 1/n2. The changes in the effect on the future can add up over n buyers to change the probability of reaching a particular buyer by at most 1/n,so we can neglect this change.

5.4.2 A Property of Poisson Distribution

The proof of Lemma 5.2.2 uses the following property of Poisson variables.

Lemma 5.4.1. Let P be a Poisson variable with mean K, i.e. for all integers m 0, ≥ m K+1 Pr [P = m]= K . Then E[max 0, P K ]= K ,andsoE[min P, K ]=E[P m!eK { − } K!eK { } − K+1 max 0, P K ]=K K . { − } − K!eK

Km(m K) KK+1 Proof. All we need to show is that m∞=K+1 m!− = K! . It is easy to show by induction that for any j 1, ≥ K+j KK+1 Km(m K) KK+j+1 − = . K! − m! (K + j)! m=K+1 Kx+1 Since limx = 0, the proof is complete. →∞ x!

134 Chapter 6

Market Making and Mean Reversion

In this chapter, we analyze the factors that affect the profits of a market maker for a financial instrument. This work is done in collaboration with Michael Kearns [25]. A market maker is a firm, individual or trading strategy that always or often quotes both a buy and a sell price for a financial instrument or commodity, hoping to make a profit by exploiting the difference between the two prices, known as the spread.Intuitively,a market maker wishes to buy and sell equal volumes of the instrument (or commodity), and thus rarely or never accumulate a large net position, and profit from the difference between the selling and buying prices. Historically, the chief purpose of market makers has been to provide liquidity to the market — the financial instrument can always be bought from, or sold to, the market maker at the quoted prices. Market makers are common in foreign exchange trading, where most trading firms offer both buying and selling rates for a currency. They also play a major role in stock exchanges, and historically exchanges have often appointed trading firms to act as official market makers for specific equities. NYSE designates a single market maker for each stock, known as the specialist for that stock. In contrast, NASDAQ allows several market makers for each stock. More recently, fast electronic trading systems have led trading firms to behave like market makers without formally being designated so. In other words, many trading firms attempt to buy and sell a stock simultaneously, and profit from the difference

135 between buying and selling prices. We shall refer to such trading algorithms generally as market making algorithms. In this chapter, we analyze the profitability of market making algorithms. Market making has existed as a trading practice for a long time, and it has also inspired significant amount of empirical as well as theoretical research [56, 52, 62, 30, 38, 39]. Most of the theoretical models [52, 56, 38, 39] view market makers as dealers who single-handedly create the market by offering buying and selling prices, and there is no trading in their absence (that is, all trades must have the market marker as one of the parties). On the other hand, much of the empirical work has focused on analyzing the behavior of specialist market makers in NYSE, using historical trading data from NYSE [62, 30]. In contrast, our theoretical and empirical work studies the behavior of market making algorithms in both very general and certain specific price time series models, where trading occurs at varying prices even in the absence of the market maker. This view seems appropriate in modern electronic markets, where any trading party whatsoever is free to quote on both sides of the market, and officially designated market makers and specialists are of diminishing importance.

Market Making vs. Statistical Arbitrage Before describing our models and results, we first offer some clarifying comments on the technical and historical differences between market making and statistical arbitrage, the latter referring to the activity of using computation-intensive quantitative modeling to design profitable automated trading strategies. Such clarification is especially called for in light of the blurred distinction between traditional market-makers and other kinds of trading activity that electronic markets have made possible, and the fact that many quantitative hedge funds that engage in statistical arbitrage may indeed have strategies that have market making behaviors. Perhaps the hallmark of market making is the willingness to always quote competitive buy and sell prices, but with the goal of minimizing directional risk. By this we mean that the market maker is averse to acquiring a large net long or short position in a stock, since in doing so there is the risk of large losses should the price move in the wrong direction. Thus if a market maker begins to acquire a large net long position, it would continue to quote a buy price, but perhaps a somewhat lower one which is less likely to get executed. Alternatively (or in addition), the strategy might choose to lower its sell quote in order to

136 increase the chances of acquiring short trades to offset its net long inventory. In this sense, pure market making strategies have no “view” or “opinion” on which direction the price “should” move — indeed, as we shall show, the most profitable scenario for a market maker is one in which there is virtually no overall directional movement of the stock, but rather a large amount of non-directional volatility. In contrast, many statistical arbitrage strategies are the opposite of market making in that they deliberately want to make directional bets — that is, they want to acquire large net positions because they have a prediction or model of future price movement. To give one classic example, in the simplest form of pairs trading, one follows the prices of two presumably related stocks, such as Microsoft and Apple. After normalizing the prices by their historical means and variances, one waits for there to be a significant gap in their current prices — for instance, Apple shares becoming quite expensive relative to Microsoft shares given the historical normalization. At this point, one takes a large short position in Apple and an offsetting large long position in Microsoft. This amounts to a bet that the prices of the two stocks will eventually return to their historical relationship: if Apple’s share price falls and Microsoft’s rises, both positions pay off. If the gap continues to grow, the strategy incurs a loss. If both rise or both fall without changing the gap between, there is neither gain nor loss. The important point here is that, in contrast to market making, the source of profitability (or loss) are directional bets rather than price volatility.

Theoretical Model and Results We ﬁrst summarize our theoretical models and our three main theoretical results. We assume there is an exogenous market where a stock can be bought and sold at prices dictated by a given time series process. At any given point of time, there is a single exogenous asset price at which the stock can both be bought as well as sold. The price evolution in the market is captured by the time series. The market making algorithm is an online decision process that can place buy and sell limit orders with some quoted limit order prices at any time, and may also cancel these orders at any future time. For simplicity, we assume that each order requests only one share of the stock (a trader may place multiple orders at the same price). If at any time after placing the order and before its cancellation, the asset price of the stock equals or exceeds (respectively, falls below) the quoted price on a sell order (respectively, buy order), then the order gets executed at the

137 quoted price, i.e. the trader pays (respectively, gains) one share and gains (respectively, pays) money equal to the price quoted on the order. We shall refer to the net volume of the stock held by a trader at a given point of time as inventory. Note that inventory may be positive (net long position) or negative (net short position). To evaluate the profit made by a market making algorithm, we shall fix a time horizon when the algorithm must liquidate its inventory at the current asset price. Our first and most general theoretical result is a succinct and exact characterization of the profit obtained by a simple market-making algorithm, given any asset price time series, in the model above. If the sum of absolute values of all local price movements (defined below) is K, and the difference between opening and closing prices is z,weshow that the profit obtained is exactly (K z2)/2. The positive term K can be viewed as the − product of the average volatility of the price and the duration for which the algorithm is run. The negative term z2 captures the net change in price during the entire trading period. Thus this characterization indicates that market making is profitable when there is a large amount of local price movement, but only a small net change in the price. This observation matches a common intuition among market makers, and provides a theoretical foundation for such a belief. An unbiased random walk (or Brownian motion) provides a boundary of profitability — the algorithm makes zero expected profit (as do all trading algorithms), while any stochastic price process whose closing price has comparatively less variance from the opening price makes positive expected profit. The last observation leads to our second result.

Mean Reversion We next exhibit the benefit of obtaining a succinct and exact expression for profit by applying it to some classes of stochastic time series that help in understanding the circumstances under which the algorithm is profitable. We identify a natural class of time series called mean-reverting processes whose properties make our market making algorithm profitable in expectation. A stochastic price series is considered to be reverting towards its long-term mean µ if the price shows a downward trend when greater than µ and upward trend when less than µ. Prices of commodities such as oil [72, 79] and foreign exchange rates [55] have been empirically observed to exhibit mean reversion. Mean- reverting stochastic processes are studied as a major class of price models, as a contrast to

138 stochastic processes with directional drift, or with no drift, such as Brownian motion. One widely studied mean-reverting stochastic process is the Ornstein-Uhlenbeck process [54]. Formally, our second result states that out market making algorithm has expected positive profit on any random walk that reverts towards its opening price. This result is quite revealing — it holds if the random walk shows even the slightest mean reversion, regardless of how complex the process may be (for instance, its evolution may depend not only on the current price, but also on the historical prices in an arbitrary way, as well as the current time). It identifies mean reversion as the natural property which renders market making profitable. Our third result shows that simple market making algorithms yield stronger profit guarantees for specific mean-reverting processes. As an example, we consider the Ornstein- Uhlenbeck (OU) process. If the price series follows this process, we show that a simple market making algorithm is profitable when run for a sufficiently long time. Moreover, the profit grows linearly with the duration for which the algorithm is run, and the profit guarantees hold not only in expectation, but with high probability. We prove this by showing that while E[K] grows linearly with time, E[z2] is upper bounded by a constant. Unlike our second result, we do not need the assumption that the price series begins at the long-term mean — the initial price appears in the upper bound on E[z2]. We also show an analogous result for another mean reverting process that has been studied in the finance literature, a model studied by Schwartz [81]. In this model, the local volatility is a linear function of price, while the OU process models volatility as a constant.

Trading Frequency: Simulations We remark that the results outlined above assume a model where the market maker can place and cancel orders as frequently as it likes, and in fact our algorithm does so after every change in the exogenous asset price. In practice, however, a market maker cannot react to every change, since the asset price may change with every trade in the market (which may or may not involve this particular market maker), and the market maker may not be able to place new limit orders after every trade of a rapidly traded stock. So we also analyze the proﬁtability of our market making algorithm when it is allowed to change its orders only after every L steps, by simulating our algorithm on random samples from the OU process. If the price series is the OU process, we show

139 that the expected proﬁt continues to grow linearly with time.

Other Related Work To our knowledge, no previous work has studied market making in an exogenously specified price time series model. Most of the theoretical work, as mentioned before, considers a single dealer model where all trades occurred through the market maker at its quoted prices [52, 56, 38, 39]. This includes the well-known Glosten-Milgrom model for market making [56]. On the other hand, there has been a fair amount of work in algorithmic trading, especially statistical arbitrage, that assumes an exogenous price time series. The closest line of research to our work in this literature is the analysis of pair trading strategies under the assumption that the price difference between the two equities show mean reversion (e.g. [47, 75]). As discussed before, such strategies are qualitatively very different from market making strategies. Moreover, most algorithmic trading work, to our knowledge, either analyze price series given by very specific stochastic differential equations (similar to Sections 6.2.1 and 6.2.2 of this chapter), or empirically analyze these algorithms against historical trading data (e.g. [53]). In contrast, we also give profit guarantees for the weakest of mean reversion processes without assuming a specific form (Theorem 18), and in fact derive an exact expression for arbitrary price series (Theorem 17), inspired from the notion of worst-case analysis in theoretical computer science.

6.1 A General Characterization

We ﬁrst describe our theoretical model formally. We assume that all events occur at discrete time steps 0, 1, 2 ...T,whereT is the time horizon when the market making algorithm must terminate. There is an asset price P of the stock at every time step 0 t T .Thus t ≤ ≤ P0,P1 ...PT is the asset price time series. We assume that all prices are integral multiples of a basic unit of money (regulations in NYSE/NASDAQ currently require prices to be integral multiples of a penny). A trading algorithm may place and cancel (possibly both) buy and sell orders at any of these time steps, and each order requests a single share at a quoted limit order price Y .A buy (respectively, sell) order at price Y placed at time t gets executed at the earliest time

140 t >tsuch that P Y (respectively, P Y ), provided that the order is not canceled t ≤ t ≥ before t. Any buy order placed by our algorithms will quote a price Y

Market Making Algorithms The basic class of market making algorithms that we consider is the following: At time t, the algorithm cancels all unexecuted orders, and places new buy orders at prices Y ,Y 1,Y 2 ...Y C and new sell orders at prices X ,X + t t − t − t − t t t 1,Xt +2...Xt + Ct,whereYt

141 only if X = X or Y = Y ). Thus the algorithm is determined by the choices of X , Y t+1 t t+1 t t t and Ct for all t, and these choices may be made after observing the price movements up to time t. We begin by presenting our basic result for a simple market making algorithm, that sets X = P + 1 and Y = P 1. t t t t − T Theorem 17. Let P0,P1 ...PT be an asset price time series. Let K = t=1 Pt Pt 1 , | − − | and let z = PT P0.Supposethat Pt+1 Pt Dt t, where Dt is known to the algorithm − | − | ≤ ∀ at time t. If the market making algorithm, that sets X = P +1, Y = P 1 and ladder t t t t − depth C = D , is run on this price series, then the inventory liquidated at time T is z, t t − and the proﬁt is (K z2)/2. − Proof. Note that at any time step t>0, at least one order gets executed if the price changes. Moreover, the number of orders executed at time t 1isPt 1 Pt (a negative ≥ − − value indicates that shares were sold). The statement holds as long as Pt 1 Pt Ct 1, | − − | ≤ − which is true by assumption. Thus K is equal to the total number of orders that gets executed. Moreover, the size of inventory held by the algorithm at time t is P P .We 0 − t shall construct disjoint pairs of all but z of the executed orders, such that each pair of | | executions comprises an executed buy and a sell execution, and the price of the buy order is 1 less than the price of the sell order, so that each such pair can be viewed as giving a proﬁt of 1.

For p>P0, we pair each sell order, priced at p, that gets executed when price increases to p or more, with the executed buy order, priced at p 1, that gets executed at the earliest − time in the future when the price falls back to p 1 or less (if the price ever falls to p 1 − − again). Note that these pairs are disjoint, since between every rise of the price from p 1 − to p, the price must obviously fall to p 1. Similarly, for p0, then the | | only executions that remain unmatched are the sell orders executed when the price increases to p and never again falls below p: for each P +z p>P , there is one such executed order. 0 ≥ 0 During liquidation at time T , these unmatched sell orders are matched by buying z shares

142 /)00 ,-. +%&'( 1!!!!!!1 23 1 !!"#$%&'()* 3

Figure 6.1: Proof by picture: Matched and unmatched trades

P0+z at price PT = P0 +z. The total loss during liquidation is ((P0 +z) p)=z(z 1)/2. p=P0+1 − − Since there are K z paired executions, the proﬁt obtained from them is (K z)/2. Hence − − the net proﬁt is (K z z(z 1))/2=(K z2)/2. A symmetric argument holds for − − − − z<0.

Note that it is typically reasonable to assume that P P

6.2 Mean Reversion Models

In this section, we use Theorem 17 to relate proﬁtability to mean reversion.

Deﬁnition 6.2.1. A unit-step walk is a series P ,P ...P such that P P 1 T> 0 1 T | t+1 − t| ≤ ∀ t 0. A stochastic price series P ,P ...P is called a random walk if it is a unit-step walk ≥ 0 1 T almost surely. We say that a random walk is unbiased if Pr [Pt+1 Pt =1Pt,Pt 1 ...P0]= − | − 1/2, for all unit-step walks P ...P ,forallT>t 0. t 0 ≥ We say a random walk is mean-reverting towards µ if

Pr [Pt+1 Pt =1Pt = x, Pt 1 ...P0] − | − ≥

Pr [Pt+1 Pt = 1 Pt = x, Pt 1 ...P0] − − | −

143 for all x µ,and ≤

Pr [Pt+1 Pt =1Pt = y, Pt 1 ...P0] − | − ≤

Pr [Pt+1 Pt = 1 Pt = x, Pt 1 ...P0] − − | − for all y µ,forallt, Pt 1,Pt 2 ...P0 such that P0,...Pt is a unit-step walk, and at least ≥ − − one of these inequalities for t

Note that all trading algorithms yield zero expected proﬁt on an unbiased random walk.

This is because the proﬁt Ft of the algorithm, if its inventory were liquidated at time t,is a martingale, irrespective of the number of shares bought or sold at each time step, and so the expected proﬁt is E[FT]=E[F0] = 0.

Theorem 18. For any random walk P0,P1 ...PT that is mean-reverting towards µ = P0, the expected profit of the market making algorithm that sets X = P +1 and Y = P 1 t t t t − (any C 0 suffices) is positive. t ≥ Proof. Since the price does not change by more than 1 in a time step, the market making algorithm need not set a ladder of prices. By Theorem 17, the expected profit is E[(K − 2 t z )/2]. Let Kt = i=1 Pi Pi 1 , and let zt = Pt P0. We show by induction on t | − − | − 2 that E[Kt] E[z ] for all t. For t = T , this would imply positive expected profit for our ≥ t algorithm. Without loss of generality, we assume that P0 = µ = 0.

For t = 0, the statement is trivially true, since Kt = zt = 0. Suppose it is true for some t 0, then we can show that it is true for t + 1. Let Ft denote the set of all unit-step ≥ walks such that P0 = µ = 0. For s Ft,letα(s)=Pr [Pt+1 Pt =1Pt,Pt 1 ...P0], and ∈ − | − let β(s)=Pr [Pt+1 Pt = 1 Pt,Pt 1 ...P0]. Also, let Pr [s] denote the probability that − − | − the ﬁrst t steps of this random walk is s.Thenwehave

E[K ]=E[K + P P ] t+1 t | t+1 − t| (6.1) = E[Kt]+ Pr [s] (α(s) + β(s)) s Ft ∈

144 2 2 E[zt+1]=E[Pt+1] = Pr [s] α(s)(P + 1)2 + β(s)(P 1)2 t t − s Ft ∈ +(1 α(s) β(s))P 2 − − t = Pr [s] P 2 + α(s)+β(s)+2P (α(s) β(s)) t t − s Ft ∈ 2 = E[Pt ]+ Pr [s] (α(s) + β(s)) s Ft ∈ +2 Pr [s] P (α(s) β(s)) t − s Ft ∈ E[K ]+ Pr [s] (α(s) + β(s)) ≤ t s Ft ∈ +2 Pr [s] P (α(s) β(s)) t − s Ft ∈ (by induction hypothesis)

= E[K ]+2 Pr [s] P (α(s) β(s)) t+1 t − s Ft ∈ (by Equation 6.1)

It suﬃces to show that that Pt(α(s) β(s)) 0 for all s Ft. This follows immediately − ≤ ∈ from the deﬁnition of a mean-reverting random walk: if Pt >P0 = 0, then α(s) < β(s), and if Pt β(s). Thus we have proved the induction hypothesis for t + 1.

Finally, for the smallest t such that for some s Ft we have α(s) = β(s), the inequality ∈ 2 in the induction hypothesis becomes strict at t + 1, i.e. E[Kt+1] > E[zt+1], and so the expected proﬁt for T>tis strictly positive.

6.2.1 Ornstein-Uhlenbeck Processes

One well-studied mean-reverting process is a continuous time, real-valued stochastic process known as the Ornstein-Uhlenbeck (OU) process [54]. We denote this process by Qt.Itis usually expressed by the following stochastic diﬀerential equation:

dQ = γ(Q µ)dt + σdW , t − t − t

145 where Wt is a standard Brownian motion, and γ, σ are positive constants, and γ < 1. The value µ is a constant around which the price ﬂuctuates — it is called the long term mean of the process. The coeﬃcient of dt is called drift, while that of dWt is called volatility.

Observe that the drift is negative for Pt >µand positive for Pt <µ— this is why the process tends to revert towards µ whenever it is far from it. γ is the rate of mean reversion.

The OU process is memoryless (distribution of Qt given Q0 is the same as distribution of

Qt+x given Qx), and given an opening value Q0, the variable Qt is known to be normally distributed, such that

γt E[Q ]=µ +(Q µ)e− , t 0 − 2 (6.2) σ 2γt Var[Q ]= (1 e− ) t 2γ −

Now we consider the OU process Qt as a price series in the unique model, and analyze profitability of our algorithm. However, since the OU process is a continuous time real- valued process, we need to define a natural restriction to a discrete integral time series that conforms to our theoretical model. We achieve this by letting Pt to be the nearest integer to Qt, for all non-negative integers t. The rounding is practical since in reality prices are not allowed to be real-valued, and further, our algorithm reacts to only integral changes in price. We shall analyze our algorithm on Pt. A significant hindrance in applying Theorem 17 to the OU process is that the jumps P P are not necessarily bounded by some constant C, so we have to put some effort | t+1 − t| into determining Ct. Since the OU process is memoryless, Equation 6.2 implies that given Q , Q is normally distributed with expectation µ +(Q µ)e γ and variance less than t t+1 t − − 2 σ2/2γ,soifwesetC>>σ √ln T and then set C = E[ Q Q Q ]+C,thenthe 2γ t | t+1 − t|| t probability that the price jump at any time exceeds the depth of the ladder is vanishing, and such events do not contribute significantly to the expected profit if we simply stop the algorithm when such an event occurs.

Theorem 19. Let P ,P ...P be a price series obtained from an OU process Q with 0 1 T { t} long-term mean µ. Then the market making algorithm that sets X = P +1, Y = P 1 t t t t − 2 and C = E[ Q Q Q ] + 10 σ √ln T yields long term expected proﬁt Ω(σT σ2/2γ t | t+1 − t|| t 2γ − − (µ Q )2). − 0

146 Proof. It is easy to show, as outlined above, that the contribution of events where a price

jump larger than Ct occurs to the expected proﬁt is negligible. We restrict our attention to the event when no such large jump occurs. By Theorem 17, the proﬁt on a sample series is 2 T (K z )/2, where K = t=1 Pt Pt 1 , and z = PT P0. The result follows by giving a − | − − | − lower bound on E[K] and an upper bound on E[z2]. Let us derive a lower bound on E[ Q Q ]. Note that this quantity is equal to | t+1 − t| E[ Q Q Q =Q], where Q is an identical but independent OU process. This is | 1 − 0 | 0 t t because the OU process is Markov, and future prices depend only on the current price. Since γ < 1, so Equation 6.2 implies that given Qt, Qt+1 is normally distributed with 2 1 e 2γ variance greater than σ /4, since − − > 1/4 when γ < 1. Hence E[ Q Q ] is at least 2γ | t+1 − t| σ/4 (using properties of a folded normal distribution). Since Pt is obtained by rounding Q ,wehave P P > Q Q 2. Thus for large enough σ (see comments at the t | t+1 − t| | t+1 − t| − end of the theorem), we get that E[K] = Ω(σT). E[z2] is approximated well enough by E[(Q Q )2], since z (Q Q ) < 2 for T − 0 | − T − 0 | all possible realizations. Again, Equation 6.2 implies that Q Q has mean µ +(Q T − 0 0 − µ)e γT Q =(µ Q )(1 e γT ) and variance σ2(1 e 2γT )/2γ.Thus,wehave − − 0 − 0 − − − − E[(Q Q )2]=Var[Q Q ]+E[Q Q ]2 T − 0 T − 0 T − 0 σ2(1 e 2γT ) = − − 2γ 2 γT 2 +(µ Q ) (1 e− ) − 0 − σ2 < +(µ Q )2 2γ − 0 Thus, E[K] grows linearly with T ,whileE[z2] is bounded by a constant. This completes the proof.

A few points worth noting about Theorem 19: our lower bound on E[K] is actually ( σ 2)T and σ must exceed 8 for this term to be positive and grow linearly with T .Thisis 4 − just an easy way to handle the arbitrary integral rounding of Qt. Intuitively, the algorithm typically cannot place orders with prices separated by less than a penny. Thus the volatility needs to be suﬃcient for integral changes to occur in the price series. If the unit of money could be made smaller, the loss due to rounding should diminish. Then σ in terms of the

new unit increases linearly, while γ remains constant (Qt becomes cQt for some scaling

147 factor c). Thus for any constant σ,asufficiently small granularity of prices allows us to apply Theorem 19. In fact, it is not difficult to see from the above analysis that the profit will grow linearly with time as long as the limiting variance of the process σ2/2γ is larger than (or comparable to) the granularity of bidding (normalized to 1). If this does not hold, then the algorithm will rarely get its orders executed, and will neither profit nor lose any significant amount. Moreover, in the proof of Theorem 19, one may note that z is a normal variable with variance bounded by a constant, while the lower bound on K grows linearly with T , and occurs with high probability. Thus the profit expression in Theorem 19 not only holds in expectation (as is the case in Theorem 18 for general mean-reverting walks), but with high probability. Furthermore, for even the smallest of γ, Theorem 19 says that the profit is positive if T is large enough. Thus, even when mean reversion is weak, a sufficiently long time horizon can make market making profitable. Finally, the profit expression of the OU process has a term (µ Q )2. While we can treat − 0 it as a constant independent of the time horizon, we can also apply another trick to reduce this constant loss if Q0 is far from µ. This is because Equation 6.2 tells us that the process converges exponentially fast towards µ — in time t = γ 1 log Q µ , E[Q µ] is down − | 0 − | | t − | to 1, and Var[Q µ] is less than σ2/γ. Thus if the horizon T is large enough, then the 0 − market maker would like to simply sit out until time t (if allowed by market regulations), and then start applying our market making strategy.

6.2.2 The Schwartz Model

We now analyze another stochastic mean reversion model that has been studied in the ﬁnance literature and was studied by Schwartz [81]. The OU process assumes that the volatility of the price curve is a constant. Schwartz proposed a model where the volatility is a linear function of the price:

dQ = γQ (ln Q ln µ)dt + σQ dW , t − t t − t t where µ is the long term mean price of the process, γ < 1, and σ < 1. Also assume that

Q0 > 0.

148 We shall show that the proﬁtability of our market making algorithm for the Schwartz model is analogous to Theorem 19: E[K] grows linearly in T , while the expected loss due to liquidation E[z2] is bounded by a constant, and hence the expected proﬁt grows linearly in T . Applying Ito’s lemma, Schwartz showed that log Qt is an OU process, and so Qt has a lognormal distribution, such that

2 σ γt γt α = E[log Q ]=(lnµ )(1 e− )+Q e− , t t − 2γ − 0 2 2 σ 2γt β = Var[log Q ]= (1 e− ) t t 2γ − Then, by properties of lognormal distributions, we have

2 2 2 E[Q ]=eαt+βt /2, Var[Q ]=(eβt 1)e2αt+βt t t − σ2 Suppose the unit of price is small enough so that ln µ> 2γ (again, this is essentially equivalent to choosing a finer granularity of placing orders). Note that shrinking the size of a unit step by a factor c leaves both σ and γ unchanged, but inflates µ by c.Sinceαt and 2 βt are upper bounded by constants, so are E[Qt] and Var[Qt], and hence E[z ] is bounded by a constant. It remains to show that E[K] = Ω(T). It suffices to show that E[ Q Q ] is at least | t+1 − t| some constant. Note that Qt is always positive, since it has a lognormal distribution. We shall show that E[ Q Q Q ] is at least some constant, for any positive Q .SinceQ | t+1 − t| t t t is a Markov process, this is equal to E[ Q Q Q =Q] for an identical but independent | 1 − 0 | 0 t 2 process Q . Observe that α (ln µ σ )(1 e γ) > 0, if Q > 0. Also, we have β2 > σ2/4. t 1 ≥ − 2γ − − 0 1 Thus σ2/4 σ2/4 2 σ2/4 Var[Q ] >e (e 1) > σ e /4 . 1 −

This shows that Var[Q1 ], given Q0, is lower bounded by a constant (that depends on σ). Since Q has a lognormal distribution, it follows that E[ Q Q ] is also lower bounded 1 | 1 − 0 | by a constant. This completes the proof for E[K] = Ω(T).

149 6.3 Trading Frequency

Our price series model makes the assumption that the market maker can place fresh orders after every change in price. In practice, however, there are many traders, and each trade causes some change in price, and an individual trader cannot react immediately to every change. We thus consider a more general model where the market maker places fresh orders after every L steps. Let us consider the same market making algorithm as before, in this infrequent order model. Thus, for every i, at time iL the algorithm places orders around

PiL in a ladder fashion as before. These orders remain unchanged (or get executed if the requisite price is reached) until time (i+1)L, and then the algorithm cancels the unexecuted orders and places fresh orders. We say that L is the trading frequency of the algorithm. The profit of the algorithm can no longer be captured succinctly as before. In particular, the profit is not exclusively determined by (nor can it be lower bounded by a function of) the prices P0,PL,P2L ...PiL ... at which the algorithm refreshes its orders — it depends on the path taken within every interval of L steps and not just the net change within this interval. Still, some of our profit guarantees continue to hold qualitatively in this model. In particular, we simulate the OU process and run our algorithm on this process, to analyze how trading frequency affects the profit of the algorithm. We simulate an OU process with γ =0.1, σ = 1 and the initial price equal to the long term mean. First, we find that the profit still shows a trend of growing linearly with time, for different trading frequencies L that are still significantly smaller than the time horizon T . We simulate the algorithm with different time horizons T and different trading frequencies, and all of them show a strong linear growth (see Figure 2). Also, the profit is expected to fall as the trading frequency increases (keeping time horizon fixed), since the number of trades executed will clearly decrease. We find that for a large enough horizon (T = 1000), this is indeed the case, but the decrease in profit is quite slow, and even with trading frequency as high as 40, the expected profit is more than 80% of the expected profit with unit trading frequency (see Figure 3). We computed average profit by simulating each setting 10000 times, to get a very narrow confidence interval. In fact, the standard deviation of the profit never exceeds 50 for any of our simulations, so the confidence interval (taken as 2σ divided by the square root of

150 400

300

200

Average Profit 100

0 200 400 600 800 1000 Time Horizon T

Figure 6.2: Proﬁt increases linearly with the time horizon, for diﬀerent trading frequencies 1, 2, 5, 10, 20.

350

300

250

200

150 Mean Profit 100

0 0 10 20 30 40 Trading Frequency

Figure 6.3: Mean proﬁt decreases slowly with trading frequency (Horizon T = 1000).

10 Standard Deviation of Profit 0 0 10 20 30 40 Trading Frequency

Figure 6.4: Standard deviation of proﬁt increases quickly with trading frequency, then stabilizes (Horizon T = 1000).

151 sample size) is less than 1, while the expected profit is much higher in all the cases. The standard deviation in the profit itself goes up sharply as the trading frequency is increased from 1, but then quickly stabilizes (see Figure 4).The increase in variance of profit can perhaps be explained by the increase in variance of the number of shares that are liquidated at the end.

6.4 Conclusion

In this chapter, we analyzed the profitability of simple market making algorithms. Market making algorithms are a restricted class of trading algorithms, though there is no formal specification of the restrictions. Intuitively, the restriction is that a market maker has to be always present in the market, wiling to buy as well as sell, and offer prices that are close to the asset price. A future direction would be to formalize such constraints, and design optimal market making algorithms that satisfy the formal restrictions.

152 Chapter 7

Summary of Results

In this chapter we revisit and summarize the results in this thesis. We brieﬂy contrast them with prior existing literature, to outline the speciﬁc contributions of this thesis to the area of computational economics.

7.1 Behavioral Study of Networked Bargaining

In Chapter 2, we conducted human subject experiments on networked bargaining, and studied the effects of network topology on bargaining power, the efficiency achieved by human subjects, and the effect of distinguishing behavioral traits such as patience or stubborn- ness on the success of individuals. Our work was inspired by a long line of theoretical work modeling and characterizing fair and/or rational outcomes in networked bargaining [51, 74, 31, 83, 17, 68, 4, 34, 27, 73, 1, 2], and the conspicuous sparsity of experimental work on the topic (except a few experiments on small networks, with no more than 6 nodes [32, 33, 83]). We conducted our experiments on fairly large networks with 36 nodes, that allowed us to test several hypotheses that could not have been tested with small networks, such as the non-local effects of network structural properties. Efficiency and effects of behavioral traits had not been studied in the previous experiments. Further, our dataset is several magnitudes larger than the previous experiments, so we were able to resolve most of our hypotheses with strong statistical significance.

153 7.2 Networked Bargaining with Non-Linear Utilities

In Chapter 3, we studied networked bargaining in a game-theoretic model, and analyzed how diminishing marginal utility among the players combine with the network topology and affect bargaining power of the nodes. Our model proposes a general method of extending any two-player cooperative bargaining solution concept (with outside options) to a networked bargaining solution concept. We studied proportional bargaining and Nash bargaining solution concepts extended to networks. Our model generalizes a linear utility model proposed by Cook and Yamagishi [31] (analyzed further by Kleinberg and Tardos [31, 4]), and can model non-linear utility functions. We showed that if players have diminishing marginal utility, it induces a rich-get-richer effect. In particular, we showed that at equilibrium, nodes with higher degree (more economic opportunities) tend to earn larger shares on each deal, even in the absence of demand and supply constraints. In contrast, if players had linear utility and no demand/supply constraint, then network structure has no effect on the shares. We also developed an algorithm that provably converges to equilibrium quickly on bipartite networks, and also characterized sufficient conditions for the existence of a unique equilibrium on regular networks.

7.3 Item Pricing in Combinatorial Auctions

In Chapter 4, we developed simple online item pricing strategies that provide the best known prior-free approximation guarantee (O(log2 mL)) on obtaining maximum revenue in combinatorial auctions with m items via any truthful mechanism, with the assumption that buyers’ valuation functions are subadditive, and that the maximum value of any item is estimated correctly up to a factor of L. Similar results were known previously for welfare [46, 43], but those strategies failed to provide good guarantees on revenue. A logarithmic guarantee to revenue was also previously known for subadditive valuations, if there were an inﬁnite supply of each distinct item [8]. A bundle pricing mechanism, developed independently by Dobzinski [42], achieves the same approximation guarantee as ours. Further, for general subadditive valuations, no mechanism is known to provide a better guarantee even with Bayesian information (constant approximation is known for two special cases, of

154 unit-demand buyers [28] and budgeted additive buyers [13] respectively).

7.4 Sequential Posted Pricing in Multi-Unit Auctions

In Chapter 5, we developed eﬃciently computable sequential pricing strategies for multi- unit auctions in a standard Bayesian setting. We showed that the ratio between the revenue of an optimal sequential pricing scheme and Myerson’s expected-revenue-optimal auction [76] tends to 1 as the size of the seller’s inventory increases, even if the valuation distributions are constructed adversarially. Earlier work [28] had established that the ratio is at least (1 1/e) for any multi-unit auction, and also constructed an instance, with a single − item to sell, to show that the ratio can be as low as 4/5. Our main contribution was to recognize that the instances with large gap can occur only when the seller is selling one or at most a few items. This result was proved independently by Yan [86]. We also developed polynomial time approximation schemes for optimal sequential posted pricing, both adaptive and non-adaptive, for multi-unit auctions. In general, analysis of adaptive policies in complex stochastic optimization problems is quite rare. For example, approximation scheme for adaptive policy was recently developed for stochastic knapsack [12], and no such result is known for slightly more general problems such as the multiple knapsack problem.

7.5 Market Making and Mean Reversion

In Chapter 6, we developed a simple trading algorithm for a financial market maker, proposed a price time series-based market model, and analyzed the performance of our algorithm in our model to show that the trading strategy is profitable when the market price exhibits mean reversion properties. The profit is affected positively by price volatility, and adversely affected by consistent directional change in the price of the asset. Our work pi- oneered the study of market making in price time series models, and is also the first one to mathematically obtain a connection between mean reversion and profitability of market making. Market making strategies have been studied previously in market models with static demand [56, 52, 38, 39], while other classes of trading strategies, such as pair trading, have been studied in price time series models [47, 75].

155 Chapter 8

Future Research Directions

This thesis makes significant progress in the understanding of large economic systems. In this final chapter, we describe a few work in progress, open problems and future research directions that are closely related to the topics addressed in this thesis, and will significantly further our understanding.

8.1 Networked Bargaining with Incomplete Information

In this work, we studied networked economic interactions with complete information (Chap- ters 2 and 3), and single-seller-multiple-buyers models with incomplete information (Chap- ters 4 and 5). A major direction forward is to study networked economic interactions with incomplete information, through theoretical modeling as well as behavioral experiments. Specifically, for every economic opportunity, a seller should know the cost, buyer should know the valuation, but the surplus is not mutually known. One specific theoretical model is to have multiple sellers choosing from the action space of truthful mechanisms. Behavioral experiments can explore how a seller learns about buyers’ (sellers’) valuations (costs), by communicating with many agents during a single game. It will be interesting to study the structure of equilibrium in this setting, the effect of network structure on the revenue of individual sellers as well as social efficiency.

156 8.2 Bundle Pricing for Multi-Parameter Mechanism Design

In our study of combinatorial auctions in Chapter 4, all our pricing strategies placed a price on each item, and the price of a bundle of items was simply the sum of the prices of items in the bundle. This simplicity is preferable because it is easy to implement, but has its limitations on how well it can perform. Moreover, many sellers are capable of oﬀering discounts on bundles of items. So it is important to understand how much more revenue bundle pricing can get compared to item pricing. In particular, can bundle pricing yield a constant approximation to revenue when buyers have subadditive valuations, even in a Bayesian setting? It is known that item pricing cannot achieve such a result, but bundle pricing may still achieve it.

8.3 Mechanism Design for Sellers with Non-Linear Utility

In our theoretical study of networked bargaining in Chapter 3, we analyzed the effect of diminishing marginal utility, that is, a concave utility function. However, this effect is not well-understood even for the single seller setting when buyers’ valuations are unknown to the seller. Myerson’s optimal mechanism characterization holds true only when the seller wishes to maximize the expected revenue. However, if the seller wishes to maximize expected utility, where utility is a non-linear function of revenue, then Myerson’s auction is no longer optimal. Concavity and other structures in the utility function can also capture risk aversion on the seller’s part in a Bayesian setting. Recent work by Sundarajan and Yan [84] designs approximately optimal mechanisms for a seller with concave utility when the buyers’ valuations are assumed to be identically and independently distributed and that the distribution satisfies a monotone hazard rate condition. Even more recently, we have developed (joint work with Anand Bhalgat and Sanjeev Khanna [11] – work in progress) efficiently computable mechanisms, with constant approximation to optimal expected utility, for multi-unit auctions and the multi-parameter setting of unit-demand buyers, which is a special case of subadditive valuations.

157 8.4 Models of Financial Markets and Optimal Strategy for Market Makers

Our model for studying a market maker’s strategy in Chapter 6 assumed a market that is not affected by the individual trader’s actions, and also possesses sufficient liquidity at the market price. A market maker is also often modeled as a single dealer in a market, such as a foreign exchange counter at an airport. In this setting, it is common to assume a Bayesian model, where the probability of buying (selling) is a function of the buying (respectively, selling) price. It will be interesting to design strategies for such a dealer that maintains limited inventory at all times, as a market maker should. The strategy can be developed as an optimization problem given the probabilistic behavior of customers, or as an online learning and optimization problem where the strategy must also learn the underlying probabilistic behavior. The problem is especially interesting if the underlying value distribution of the stock changes over time. Finally, it will be interesting to consider models that take into account the reaction of the market, that is, the reaction of other players, to the market maker’s actions. Moreover, as mentioned before, the actual trading mechanism in modern equities market is very different from our theoretical model based on price time series. Most modern exchanges, including NYSE and NASDAQ, operate according to what is known as the open limit order book mechanism, where buy and sell orders must get matched to be executed. There are few theoretical models for the evolution of such a process. It will be interesting to create evolution models for this mechanism, and study the performance of market making algorithms in those models.

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